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In mathematics, the Riemann hypothesis is the
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
that the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
has its zeros only at the negative even integers and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s with
real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. Many consider it to be the most important unsolved problem in
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
. It is of great interest in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
because it implies results about the distribution of
prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with
Goldbach's conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ...
and the
twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
, make up Hilbert's eighth problem in
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
's list of twenty-three unsolved problems; it is also one of the
Clay Mathematics Institute The Clay Mathematics Institute (CMI) is a private, non-profit foundation (nonprofit), foundation dedicated to increasing and disseminating mathematics, mathematical knowledge. Formerly based in Peterborough, New Hampshire, the corporate address i ...
's
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. According ...
, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the
Riemann hypothesis for curves over finite fields In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right) where is a non-singular -dimensional projective algeb ...
. The Riemann zeta function ζ(''s'') is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
whose
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
''s'' may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(''s'') = 0 when ''s'' is one of −2, −4, −6, .... These are called its ''trivial zeros''. The zeta function is also zero for other values of ''s'', which are called ''nontrivial zeros''. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: Thus, if the hypothesis is correct, all the nontrivial zeros lie on the ''critical line'' consisting of the complex numbers where ''t'' is a
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
and ''i'' is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.


Riemann zeta function

The
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is defined for complex ''s'' with real part greater than 1 by the
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is s ...
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
:\zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
already considered this series in the 1730s for real values of s, in conjunction with his solution to the
Basel problem The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
. He also proved that it equals the
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eu ...
:\zeta(s) = \prod_ \frac= \frac\cdot\frac\cdot\frac\cdot\frac \cdot \frac \cdots where the
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of a sequence, limit of the Multiplication#Capital pi notation, partial products ''a' ...
extends over all prime numbers ''p''. The Riemann hypothesis discusses zeros outside the
region of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series c ...
of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex ''s''. Because the zeta function is
meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
, all choices of how to perform this analytic continuation will lead to the same result, by the
identity theorem In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions ''f'' and ''g'' analytic on a domain ''D'' (open and connected subset of \mathbb or \mathbb), if ''f'' = ''g'' on so ...
. A first step in this continuation observes that the series for the zeta function and the
Dirichlet eta function In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: \eta(s) = \sum_^ = \frac - \frac + \frac - \frac + \cdo ...
satisfy the relation :\left(1-\frac\right)\zeta(s) = \eta(s) = \sum_^\infty \frac = \frac - \frac + \frac - \cdots, within the region of convergence for both series. However, the zeta function series on the right converges not just when the real part of ''s'' is greater than one, but more generally whenever ''s'' has positive real part. Thus, the zeta function can be redefined as \eta(s)/(1-2/2^s), extending it from to a larger domain: , except for the points where 1-2/2^s is zero. These are the points s = 1 + 2\pi in/\log 2 where n can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see ), giving a finite value for all values of ''s'' with positive real part except for the
simple pole In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
at ''s'' = 1. In the strip this extension of the zeta function satisfies the
functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
:\zeta(s) = 2^s\pi^\ \sin\left(\frac\right)\ \Gamma(1-s)\ \zeta(1-s). One may then define ζ(''s'') for all remaining nonzero complex numbers ''s'' ( and ''s'' ≠ 0) by applying this equation outside the strip, and letting ζ(''s'') equal the right-hand side of the equation whenever ''s'' has non-positive real part (and ''s'' ≠ 0). If ''s'' is a negative even integer then ζ(''s'') = 0 because the factor sin(π''s''/2) vanishes; these are the ''trivial zeros'' of the zeta function. (If ''s'' is a positive even integer this argument does not apply because the zeros of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
function are cancelled by the poles of the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
as it takes negative integer arguments.) The value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(''s'') as ''s'' approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all non-trivial zeros lie in the ''critical strip'' where ''s'' has real part between 0 and 1.


Origin

Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes (''x'') less than or equal to a given number ''x'', which he published in his 1859 paper "
On the Number of Primes Less Than a Given Magnitude " die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is seminal9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der K ...
". His formula was given in terms of the related function :\Pi(x) = \pi(x) + \tfrac \pi(x^) +\tfrac \pi(x^) + \tfrac\pi(x^) + \tfrac \pi(x^) +\tfrac\pi(x^) +\cdots which counts the primes and prime powers up to ''x'', counting a prime power ''p''''n'' as . The number of primes can be recovered from this function by using the
Möbius inversion formula In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large gener ...
, :\begin \pi(x) &= \sum_^\frac\Pi(x^) \\ &= \Pi(x) -\frac\Pi(x^) - \frac\Pi(x^) - \frac\Pi(x^) + \frac \Pi(x^) -\cdots, \end where ''μ'' is the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
. Riemann's formula is then :\Pi_0(x) = \operatorname(x) - \sum_\rho \operatorname(x^\rho) -\log 2 + \int_x^\infty\frac where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits: :\Pi_0(x) = \lim_\frac2. The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset)
logarithmic integral function In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
given by the
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand ...
of the divergent integral :\operatorname(x) = \int_0^x\frac. The terms li(''x''''ρ'') involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for ''x'' > 1) by analytic continuation in the complex variable ''ρ'' in the region Re(''ρ'') > 0, i.e. they should be considered as . The other terms also correspond to zeros: the dominant term li(''x'') comes from the pole at ''s'' = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see or . This formula says that the zeros of the Riemann zeta function control the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line and he knew that all of its non-trivial zeros must lie in the range He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.


Consequences

The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.


Distribution of prime numbers

Riemann's explicit formula In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied ...
for the number of primes less than a given number in terms of a sum over the zeros of the Riemann zeta function says that the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular the error term in the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
is closely related to the position of the zeros. For example, if β is the
upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...
of the real parts of the zeros, then \pi(x) - \operatorname(x) = O \left( x^ \log x \right). It is already known that 1/2 ≤ β ≤ 1. Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of Koch's result, due to , says that the Riemann hypothesis implies :, \pi(x) - \operatorname(x), < \frac \sqrt \log(x), \qquad \text x \ge 2657, where \pi(x) is the
prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ...
, \operatorname(x) is the
logarithmic integral function In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a ...
, and \log(x) is the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of ''x''. also showed that the Riemann hypothesis implies :, \psi(x) - x, < \frac \sqrt \log^2 x, \qquad \text x \ge 73.2, where \psi(x) is Chebyshev's second function. proved that the Riemann hypothesis implies that for all x \geq 2 there is a prime p satisfying :x - \frac \sqrt \log x < p \leq x. This is an explicit version of a theorem of Cramér.


Growth of arithmetic functions

The Riemann hypothesis implies strong bounds on the growth of many other
arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
s, in addition to the primes counting function above. One example involves the
Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
μ. The statement that the equation :\frac = \sum_^\infty \frac is valid for every ''s'' with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the
Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive re ...
is defined by :M(x) = \sum_ \mu(n) then the claim that :M(x) = O\left(x^\right) for every positive ε is equivalent to the Riemann hypothesis (
J.E. Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Raman ...
, 1912; see for instance: paragraph 14.25 in ). (For the meaning of these symbols, see
Big O notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
.) The
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the order ''n'' Redheffer matrix is equal to ''M''(''n''), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. The Riemann hypothesis puts a rather tight bound on the growth of ''M'', since disproved the slightly stronger
Mertens conjecture In mathematics, the Mertens conjecture is the statement that the Mertens function M(n) is bounded by \pm\sqrt. Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 ...
:, M(x), \le \sqrt x. The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(''n''). A typical example is
Robin's theorem In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (includin ...
, which states that if σ(''n'') is the sigma function, given by :\sigma(n) = \sum_ d then :\sigma(n) < e^\gamma n \log \log n for all ''n'' > 5040 if and only if the Riemann hypothesis is true, where γ is the
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural l ...
. A related bound was given by
Jeffrey Lagarias Jeffrey Clark Lagarias (born November 16, 1949 in Pittsburgh, Pennsylvania, United States) is a mathematician and professor at the University of Michigan. Education While in high school in 1966, Lagarias studied astronomy at the Summer Science P ...
in 2002, who proved that the Riemann hypothesis is equivalent to the statement that: : \sigma(n) < H_n + \log(H_n)e^ for every
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
''n'' > 1, where H_n is the ''n''th
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dot ...
. The Riemann hypothesis is also true if and only if the inequality :\frac is true for all ''n'' ≥ ''p''120569# where φ(''n'') is
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
and ''p''120569# is the product of the first 120569 primes. Another example was found by
Jérôme Franel Jérôme Franel (1859–1939) was a Swiss mathematician who specialised in analytic number theory. He is mainly known through a 1924 paper, in which he establishes the equivalence of the Riemann hypothesis to a statement on the size of the discre ...
, and extended by
Landau Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
(see ). The Riemann hypothesis is equivalent to several statements showing that the terms of the
Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ...
are fairly regular. One such equivalence is as follows: if ''F''''n'' is the Farey sequence of order ''n'', beginning with 1/''n'' and up to 1/1, then the claim that for all ε > 0 :\sum_^m, F_n(i) - \tfrac, = O\left(n^\right) is equivalent to the Riemann hypothesis. Here :m = \sum_^n\phi(i) is the number of terms in the Farey sequence of order ''n''. For an example from
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, if ''g''(''n'') is Landau's function given by the maximal order of elements of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
''S''''n'' of degree ''n'', then showed that the Riemann hypothesis is equivalent to the bound :\log g(n) < \sqrt for all sufficiently large ''n''.


Lindelöf hypothesis and growth of the zeta function

The Riemann hypothesis has various weaker consequences as well; one is the
Lindelöf hypothesis In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see ) about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It s ...
on the rate of growth of the zeta function on the critical line, which says that, for any ''ε'' > 0, :\zeta\left(\frac + it\right) = O(t^\varepsilon), as t \to \infty. The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that : e^\gamma\le \limsup_\frac\le 2e^\gamma : \frace^\gamma\le \limsup_\frac\le \frace^\gamma so the growth rate of ζ(1+''it'') and its inverse would be known up to a factor of 2.


Large prime gap conjecture

The prime number theorem implies that on average, the gap between the prime ''p'' and its successor is log ''p''. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is ''O''( log ''p''). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true:
Cramér's conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and t ...
implies that every gap is ''O''((log ''p'')2), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.


Analytic criteria equivalent to the Riemann hypothesis

Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the
divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
σ(''n'').) The
Riesz criterion In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series :(x) = -\sum_^\infty \frac=x \sum_^\infty \frac \exp\left(\frac\right). If we set F(x) = \fr ...
was given by , to the effect that the bound :-\sum_^\infty \frac= O\left(x^\right) holds for all ε > 0 if and only if the Riemann hypothesis holds. proved that the Riemann hypothesis is true if and only if the space of functions of the form :f(x) = \sum_^nc_\nu\rho \left(\frac \right) where ρ(''z'') is the fractional part of ''z'', , and :\sum_^nc_\nu\theta_\nu=0, is dense in the Hilbert space ''L''2(0,1) of square-integrable functions on the unit interval. extended this by showing that the zeta function has no zeros with real part greater than 1/''p'' if and only if this function space is dense in ''Lp''(0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where \theta_\nu \in \_. showed that the Riemann hypothesis is true if and only if the integral equation :\int_^\infty\frac\,dz=0 has no non-trivial bounded solutions \phi for 1/2<\sigma <1.
Weil's criterion In mathematics, Weil's criterion is a criterion of André Weil for the Generalized Riemann hypothesis to be true. It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite. Weil's idea wa ...
is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is
Li's criterion In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. In 1999, Enrico Bombieri and J ...
, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis. proved that the Riemann hypothesis is equivalent to the statement that \zeta'(s), the derivative of \zeta(s), has no zeros in the strip :0 < \Re(s) < \frac12. That \zeta(s) has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line. The
Farey sequence In mathematics, the Farey sequence of order ''n'' is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to ''n'', arranged in ord ...
provides two equivalences, due to Jerome Franel and
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold ...
in 1924. The
De Bruijn–Newman constant The de Bruijn–Newman constant, denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is a mathematical constant defined via the zero of a function, zeros of a certain function (mathematics), function ''H''(''λ'',  ...
denoted by Λ and named after
Nicolaas Govert de Bruijn Nicolaas Govert (Dick) de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic.
and Charles M. Newman, is defined as the unique real number such that the
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
:H(\lambda, z):=\int_^ e^ \Phi(u) \cos (z u)\, d u, that is parametrised by a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
parameter ''λ'', has a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
variable ''z'' and is defined using a super-exponentially decaying function :\Phi(u) = \sum_^ (2\pi^2n^4e^ - 3 \pi n^2 e^ ) e^. has only real zeros if and only if λ ≥ Λ. Since the Riemann hypothesis is equivalent to the claim that all the zeroes of ''H''(0, ''z'') are real, the Riemann hypothesis is equivalent to the conjecture that \Lambda\leq 0. Brad Rodgers and
Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
discovered the equivalence is actually \Lambda = 0 by proving zero to be the lower bound of the constant. Proving zero is also the upper bound would therefore prove the Riemann hypothesis. As of April 2020 the upper bound is \Lambda\leq 0.2.


Consequences of the generalized Riemann hypothesis

Several applications use the
generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
for
Dirichlet L-series In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...
or zeta functions of number fields rather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily be generalized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for the Riemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Several results first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it, though these were usually much harder. Many of the consequences on the following list are taken from . * In 1913, Grönwall showed that the generalized Riemann hypothesis implies that Gauss's list of imaginary quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditional proofs of this without using the generalized Riemann hypothesis. * In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture of Chebyshev that \lim_ \sum_(-1)^ x^p=+\infty, which says that primes 3 mod 4 are more common than primes 1 mod 4 in some sense. (For related results, see .) * In 1923, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of three primes, though in 1937 Vinogradov gave an unconditional proof. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed that the generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of three primes. In 2013
Harald Helfgott Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher ('' directeur de recherche'') at the CNRS at the Institut Mathématique de Jussieu, Paris. Early life and education ...
proved the ternary Goldbach conjecture without the GRH dependence, subject to some extensive calculations completed with the help of David J. Platt. * In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmetic progression ''a'' mod ''m'' is at most ''Km''2log(''m'')2 for some fixed constant ''K''. * In 1967, Hooley showed that the generalized Riemann hypothesis implies
Artin's conjecture on primitive roots In number theory, Artin's conjecture on primitive roots states that a given integer ''a'' that is neither a square number nor −1 is a primitive root modulo infinitely many primes ''p''. The conjecture also ascribes an asymptotic density to the ...
. * In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of
idoneal number In mathematics, Leonhard Euler, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''2 ± ''Dy''2 (where ''x''2 is relativel ...
s is complete. * showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number fields implies that any number field with class number 1 is either Euclidean or an imaginary quadratic number field of
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
−19, −43, −67, or −163. * In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is prime in polynomial time via the
Miller test The Miller test, also called the three-prong obscenity test, is the United States Supreme Court's test for determining whether speech or expression can be labeled obscene, in which case it is not protected by the First Amendment to the United St ...
. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this result unconditionally using the
AKS primality test The AKS primality test (also known as Agrawal–Kayal–Saxena primality test and cyclotomic AKS test) is a deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Determi ...
. * discussed how the generalized Riemann hypothesis can be used to give sharper estimates for discriminants and class numbers of number fields. * showed that the generalized Riemann hypothesis implies that Ramanujan's integral quadratic form represents all integers that it represents locally, with exactly 18 exceptions. * In 2021, Alexander (Alex) Dunn and Maksym Radziwill proved Patterson's conjecture under the assumption of the GRH.


Excluded middle

Some consequences of the RH are also consequences of its negation, and are thus theorems. In their discussion of the Hecke, Deuring, Mordell, Heilbronn theorem, say
The method of proof here is truly amazing. If the generalized Riemann hypothesis is true, then the theorem is true. If the generalized Riemann hypothesis is false, then the theorem is true. Thus, the theorem is true!!     (punctuation in original)
Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.


Littlewood's theorem

This concerns the sign of the error in the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
. It has been computed that π(''x'') < li(''x'') for all ''x'' ≤ 1025 (see this
table Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table data ...
), and no value of ''x'' is known for which π(''x'') > li(''x''). In 1914 Littlewood proved that there are arbitrarily large values of ''x'' for which :\pi(x)>\operatorname(x) +\frac13\frac\log\log\log x, and that there are also arbitrarily large values of ''x'' for which :\pi(x)<\operatorname(x) -\frac13\frac\log\log\log x. Thus the difference π(''x'') − li(''x'') changes sign infinitely many times.
Skewes' number In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which :\pi(x) > \operatorname(x), where is the prime-counting function ...
is an estimate of the value of ''x'' corresponding to the first sign change. Littlewood's proof is divided into two cases: the RH is assumed false (about half a page of ), and the RH is assumed true (about a dozen pages). followed this up with a paper on the number of times \Delta(n) changes sign in the interval \Delta(n) .


Gauss's class number conjecture

This is the
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
(first stated in article 303 of Gauss's ''
Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
'') that there are only finitely many imaginary quadratic fields with a given class number. One way to prove it would be to show that as the discriminant the class number . The following sequence of theorems involving the Riemann hypothesis is described in : (In the work of Hecke and Heilbronn, the only ''L''-functions that occur are those attached to imaginary quadratic characters, and it is only for those ''L''-functions that ''GRH is true'' or ''GRH is false'' is intended; a failure of GRH for the ''L''-function of a cubic Dirichlet character would, strictly speaking, mean GRH is false, but that was not the kind of failure of GRH that Heilbronn had in mind, so his assumption was more restricted than simply ''GRH is false''.) In 1935, Carl Siegel later strengthened the result without using RH or GRH in any way.


Growth of Euler's totient

In 1983 J. L. Nicolas proved that \varphi(n) < e^\frac for infinitely many ''n'', where φ(''n'') is
Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
and γ is
Euler's constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natural ...
. Ribenboim remarks that: "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."


Generalizations and analogs


Dirichlet L-series and other number fields

The Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, but much more general, global
L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ris ...
s. In this broader setting, one expects the non-trivial zeros of the global ''L''-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for the single Riemann zeta function, which account for the true importance of the Riemann hypothesis in mathematics. The
generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
extends the Riemann hypothesis to all
Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...
s. In particular it implies the conjecture that
Siegel zero Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Al ...
s (zeros of ''L''-functions between 1/2 and 1) do not exist. The
extended Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
extends the Riemann hypothesis to all
Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
s of
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to the generalized Riemann hypothesis. The Riemann hypothesis can also be extended to the ''L''-functions of
Hecke character In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and ce ...
s of number fields. The
grand Riemann hypothesis In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all automorphic L-function, automorphic L-function, ''L''-functions lie on the crit ...
extends it to all automorphic zeta functions, such as
Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used i ...
s of
Hecke eigenform In mathematics, an eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators ''Tm'', ''m'' = 1, 2, 3, .... Eigenforms fall into the realm ...
s.


Function fields and zeta functions of varieties over finite fields

introduced global zeta functions of (quadratic) function fields and conjectured an analogue of the Riemann hypothesis for them, which has been proved by Hasse in the genus 1 case and by in general. For instance, the fact that the
Gauss sum In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a ...
, of the quadratic character of a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of size ''q'' (with ''q'' odd), has absolute value \sqrt is actually an instance of the Riemann hypothesis in the function field setting. This led to conjecture a similar statement for all
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
; the resulting
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...
were proved by .


Arithmetic zeta functions of arithmetic schemes and their L-factors

Arithmetic zeta function In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmeti ...
s generalise the Riemann and Dedekind zeta functions as well as the zeta functions of varieties over finite fields to every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme of Kronecker dimension ''n'' can be factorized into the product of appropriately defined L-factors and an auxiliary factor . Assuming a functional equation and meromorphic continuation, the generalized Riemann hypothesis for the L-factor states that its zeros inside the critical strip \Re(s)\in (0,n) lie on the central line. Correspondingly, the generalized Riemann hypothesis for the arithmetic zeta function of a regular connected equidimensional arithmetic scheme states that its zeros inside the critical strip lie on vertical lines \Re(s)=1/2,3/2,\dots,n-1/2 and its poles inside the critical strip lie on vertical lines \Re(s)=1, 2, \dots,n-1. This is known for schemes in positive characteristic and follows from , but remains entirely unknown in characteristic zero.


Selberg zeta functions

introduced the
Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
of a Riemann surface. These are similar to the Riemann zeta function: they have a functional equation, and an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The
Selberg trace formula In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given b ...
is the analogue for these functions of the explicit formulas in prime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis, with the imaginary parts of their zeros related to the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of the Laplacian operator of the Riemann surface.


Ihara zeta functions

The
Ihara zeta function In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first ...
of a finite graph is an analogue of the
Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
, which was first introduced by Yasutaka Ihara in the context of discrete subgroups of the two-by-two p-adic special linear group. A regular finite graph is a
Ramanujan graph In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders. AMurty's survey papernotes, Ramanu ...
, a mathematical model of efficient communication networks, if and only if its Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Sunada.


Montgomery's pair correlation conjecture

suggested the pair correlation conjecture that the correlation functions of the (suitably normalized) zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix. showed that this is supported by large-scale numerical calculations of these correlation functions. Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a related conjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linear relations between their imaginary parts).
Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
s of algebraic number fields, which generalize the Riemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorize as a product of powers of
Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory ...
s, so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.


Other zeta functions

There are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which have been proved.
Goss zeta function In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. proved that it satisfies an analogue of the Riemann hypothesis. proved results for a higher-dimensiona ...
s of function fields have a Riemann hypothesis, proved by . The main conjecture of
Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In th ...
, proved by Barry Mazur and
Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awar ...
for
cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th ...
s, and Wiles for totally real fields, identifies the zeros of a ''p''-adic ''L''-function with the eigenvalues of an operator, so can be thought of as an analogue of the
Hilbert–Pólya conjecture In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theor ...
for ''p''-adic ''L''-functions.


Attempted proofs

Several mathematicians have addressed the Riemann hypothesis, but none of their attempts has yet been accepted as a proof. lists some incorrect solutions.


Operator theory

Hilbert and Pólya suggested that one way to derive the Riemann hypothesis would be to find a
self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
, from the existence of which the statement on the real parts of the zeros of ζ(''s'') would follow when one applies the criterion on real
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s. Some support for this idea comes from several analogues of the Riemann zeta functions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over a finite field correspond to eigenvalues of a
Frobenius element In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
on an
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...
group, the zeros of a
Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
are eigenvalues of a
Laplacian operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
of a Riemann surface, and the zeros of a p-adic zeta function correspond to eigenvectors of a Galois action on
ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
s. showed that the distribution of the zeros of the Riemann zeta function shares some statistical properties with the eigenvalues of
random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...
drawn from the
Gaussian unitary ensemble In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathe ...
. This gives some support to the
Hilbert–Pólya conjecture In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theor ...
. In 1999, Michael Berry and
Jonathan Keating Jonathan Peter Keating (born 20 September 1963) is a British mathematician. As of September 2019, he is the Sedleian Professor of Natural Philosophy at the University of Oxford, and from 2012 to 2019 was the Henry Overton Wills Professor of Math ...
conjectured that there is some unknown quantization \hat H of the classical Hamiltonian ''H'' = ''xp'' so that \zeta (1/2+i\hat H) = 0 and even more strongly, that the Riemann zeros coincide with the spectrum of the operator 1/2 + i \hat H. This is in contrast to
canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible. Historically, this was not quite ...
, which leads to the
Heisenberg uncertainty principle In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
\sigma_x \sigma_p \geq \frac and the
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
as spectrum of the
quantum harmonic oscillator 量子調和振動子 は、 古典調和振動子 の 量子力学 類似物です。任意の滑らかな ポテンシャル は通常、安定した 平衡点 の近くで 調和ポテンシャル として近似できるため、最 ...
. The crucial point is that the Hamiltonian should be a self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program. In a connection with this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of the Hamiltonian is connected to the
half-derivative Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration ...
of the function N(s)= \frac\operatorname\xi(1/2+i\sqrt s) then, in Berry–Connes approach V^(x) = \sqrt \frac. This yields a Hamiltonian whose eigenvalues are the square of the imaginary part of the Riemann zeros, and also that the
functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the i ...
of this
Hamiltonian operator Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonia ...
is just the
Riemann Xi function In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. Definition Riemann's original lower-ca ...
. In fact the Riemann Xi function would be proportional to the functional determinant ( Hadamard product) \det(H+1/4+s(s-1)) as proved by Connes and others, in this approach \frac=\frac. The analogy with the Riemann hypothesis over
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s suggests that the Hilbert space containing eigenvectors corresponding to the zeros might be some sort of first
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
Spec (''Z'') of the integers. described some of the attempts to find such a cohomology theory. constructed a natural space of invariant functions on the upper half plane that has eigenvalues under the Laplacian operator that correspond to zeros of the Riemann zeta function—and remarked that in the unlikely event that one could show the existence of a suitable positive definite inner product on this space, the Riemann hypothesis would follow. discussed a related example, where due to a bizarre bug a computer program listed zeros of the Riemann zeta function as eigenvalues of the same
Laplacian operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
. surveyed some of the attempts to construct a suitable physical model related to the Riemann zeta function.


Lee–Yang theorem

The Lee–Yang theorem states that the zeros of certain partition functions in
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
all lie on a "critical line" with their real part equals to 0, and this has led to some speculation about a relationship with the Riemann hypothesis.


Turán's result

showed that if the functions \sum_^N n^ have no zeros when the real part of ''s'' is greater than one then T(x) = \sum_\frac\ge 0\text x > 0, where λ(''n'') is the
Liouville function The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...
given by (−1)''r'' if ''n'' has ''r'' prime factors. He showed that this in turn would imply that the Riemann hypothesis is true. But proved that ''T''(''x'') is negative for infinitely many ''x'' (and also disproved the closely related
Pólya conjecture In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an ''odd'' number of prime factors. The conjecture was set forth by the Hungarian mat ...
), and showed that the smallest such ''x'' is . showed by numerical calculation that the finite
Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analyti ...
above for ''N''=19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, the nonexistence of zeros with real part greater than 1+''N''−1/2+ε for large ''N'' in the finite Dirichlet series above, would also imply the Riemann hypothesis, but showed that for all sufficiently large ''N'' these series have zeros with real part greater than . Therefore, Turán's result is
vacuously true In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied. For example, the statement "she ...
and cannot help prove the Riemann hypothesis.


Noncommutative geometry

has described a relationship between the Riemann hypothesis and
noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...
, and showed that a suitable analog of the
Selberg trace formula In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given b ...
for the action of the idèle class group on the adèle class space would imply the Riemann hypothesis. Some of these ideas are elaborated in .


Hilbert spaces of entire functions

showed that the Riemann hypothesis would follow from a positivity condition on a certain
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
of
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s. However showed that the necessary positivity conditions are not satisfied. Despite this obstacle, de Branges has continued to work on an attempted proof of the Riemann hypothesis along the same lines, but this has not been widely accepted by other mathematicians.


Quasicrystals

The Riemann hypothesis implies that the zeros of the zeta function form a
quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical cr ...
, a distribution with discrete support whose
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
also has discrete support. suggested trying to prove the Riemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.


Arithmetic zeta functions of models of elliptic curves over number fields

When one goes from geometric dimension one, e.g. an
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
, to geometric dimension two, e.g. a regular model of an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
over a number field, the two-dimensional part of the generalized Riemann hypothesis for the
arithmetic zeta function In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithmeti ...
of the model deals with the poles of the zeta function. In dimension one the study of the zeta integral in
Tate's thesis In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function ...
does not lead to new important information on the Riemann hypothesis. Contrary to this, in dimension two work of Ivan Fesenko on two-dimensional generalisation of Tate's thesis includes an integral representation of a zeta integral closely related to the zeta function. In this new situation, not possible in dimension one, the poles of the zeta function can be studied via the zeta integral and associated adele groups. Related conjecture of on the positivity of the fourth derivative of a boundary function associated to the zeta integral essentially implies the pole part of the generalized Riemann hypothesis. proved that the latter, together with some technical assumptions, implies Fesenko's conjecture.


Multiple zeta functions

Deligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zeros and poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of the zeros of the original zeta function. By analogy, introduced multiple zeta functions whose zeros and poles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restricted to sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles of the multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.


Location of the zeros


Number of zeros

The functional equation combined with the
argument principle In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. Specifically, i ...
implies that the number of zeros of the zeta function with imaginary part between 0 and ''T'' is given by :N(T)=\frac\mathop(\xi(s)) = \frac\mathop(\Gamma(\tfrac)\pi^\zeta(s)s(s-1)/2) for ''s''=1/2+i''T'', where the argument is defined by varying it continuously along the line with Im(''s'')=''T'', starting with argument 0 at ∞+i''T''. This is the sum of a large but well understood term :\frac\mathop(\Gamma(\tfrac)\pi^s(s-1)/2) = \frac\log\frac-\frac +7/8+O(1/T) and a small but rather mysterious term :S(T) = \frac\mathop(\zeta(1/2+iT)) =O(\log T). So the density of zeros with imaginary part near ''T'' is about log(''T'')/2π, and the function ''S'' describes the small deviations from this. The function ''S''(''t'') jumps by 1 at each zero of the zeta function, and for it decreases monotonically between zeros with derivative close to −log ''t''. proved that, if T > e, then :, N(T) - \frac \log, \leq 0.112 \log T + 0.278 \log\log T + 3.385 + \frac. Karatsuba (1996) proved that every interval (''T'', ''T''+''H''] for H \ge T^ contains at least : H(\log T)^e^ points where the function ''S''(''t'') changes sign. showed that the average moments of even powers of ''S'' are given by :\int_0^T, S(t), ^dt = \fracT(\log \log T)^k + O(T(\log \log T)^). This suggests that ''S''(''T'')/(log log ''T'')1/2 resembles a Gaussian random variable with mean 0 and variance 2π2 ( proved this fact). In particular , ''S''(''T''), is usually somewhere around (log log ''T'')1/2, but occasionally much larger. The exact order of growth of ''S''(''T'') is not known. There has been no unconditional improvement to Riemann's original bound ''S''(''T'')=O(log ''T''), though the Riemann hypothesis implies the slightly smaller bound ''S''(''T'')=O(log ''T''/log log ''T''). The true order of magnitude may be somewhat less than this, as random functions with the same distribution as ''S''(''T'') tend to have growth of order about log(''T'')1/2. In the other direction it cannot be too small: showed that , and assuming the Riemann hypothesis Montgomery showed that . Numerical calculations confirm that ''S'' grows very slowly: , ''S''(''T''),  < 1 for , , ''S''(''T''),  < 2 for ''T'' < , and the largest value of , ''S''(''T''), found so far is not much larger than 3. Riemann's estimate ''S''(''T'') = O(log ''T'') implies that the gaps between zeros are bounded, and Littlewood improved this slightly, showing that the gaps between their imaginary parts tend to 0.


Theorem of Hadamard and de la Vallée-Poussin

and independently proved that no zeros could lie on the line Re(''s'') = 1. Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showed that all non-trivial zeros must lie in the interior of the critical strip . This was a key step in their first proofs of the
prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
. Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that if ζ(1+''it'') vanishes, then ζ(1+2''it'') is singular, which is not possible. One way of doing this is by using the inequality :, \zeta(\sigma)^3\zeta(\sigma+it)^4\zeta(\sigma+2it), \ge 1 for σ > 1, ''t'' real, and looking at the limit as σ → 1. This inequality follows by taking the real part of the log of the Euler product to see that :, \zeta(\sigma+it), = \exp\Re\sum_\frac=\exp\sum_\frac, where the sum is over all prime powers ''p''''n'', so that :, \zeta(\sigma)^3\zeta(\sigma+it)^4\zeta(\sigma+2it), = \exp\sum_p^\frac which is at least 1 because all the terms in the sum are positive, due to the inequality :3+4\cos(\theta)+\cos(2\theta) = 2 (1+\cos(\theta))^2\ge0.


Zero-free regions

The most extensive computer search by Platt and Trudgian for counter examples of the Riemann hypothesis has verified it for , t, \leq 3.0001753328 \cdot 10^ . Beyond that zero-free regions are known as inequalities concerning , which can be zeroes. The oldest version is from De la Vallée-Poussin (1899–1900), who proved there is a region without zeroes that satisfies for some positive constant ''C''. In other words, zeros cannot be too close to the line there is a zero-free region close to this line. This has been enlarged by several authors using methods such as
Vinogradov's mean-value theorem In mathematics, Vinogradov's mean value theorem is an estimate for the number of equal sums of powers. It is an important inequality in analytic number theory, named for I. M. Vinogradov. More specifically, let J_(X) count the number of solution ...
. The most recent paper by Mossinghoff, Trudgian and Yang is from December 2022 and provides four zero-free regions that improved the previous results of Kevin Ford from 2002, Mossinghoff and Trudgian themselves from 2015 and Pace Nielsen's slight improvement of Ford from October 2022: : \sigma\ge 1 - \frac whenever , t, \geq 2 , : \sigma\ge 1-\frac whenever , t, \geq 3 (largest known region in the bound 3.0001753328 \cdot 10^ \leq , t, \leq \exp(64.1) \approx 6.89 \cdot 10^ ), : \sigma\ge 1 - \frac whenever , t, \geq 1.88 \cdot 10^ (largest known region in the bound \exp(64.1) \leq , t, \leq exp(1000) \approx 1.97 \cdot 10^ ) and : \sigma\ge 1-\frac+\frac whenever , t, \geq exp(1000) (largest known region in its own bound) The paper also has a improvement to the second zero-free region, whose bounds are unknown on accound of , t, being merely assumed to be "sufficiently large" to fulfill the requirements of the paper's proof. This region is \sigma\ge 1-\frac.


Zeros on the critical line

and showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. proved that at least a (small) positive proportion of zeros lie on the line. improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and improved this further to two-fifths. In 2020, this estimate was extended to five-twelfths by Pratt, Robles, Zaharescu and Zeindler by considering extended mollifiers that can accommodate higher order derivatives of the zeta function and their associated Kloosterman sums. Most zeros lie close to the critical line. More precisely, showed that for any positive ε, the number of zeroes with real part at least 1/2+ε and imaginary part at between ''-T'' and ''T'' is O(T). Combined with the facts that zeroes on the critical strip are symmetric about the critical line and that the total number of zeroes in the critical strip is \Theta(T\log T),
almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...
non-trivial zeroes are within a distance ε of the critical line. gives several more precise versions of this result, called ''zero density estimates'', which bound the number of zeros in regions with imaginary part at most ''T'' and real part at least 1/2+ε.


Hardy–Littlewood conjectures

In 1914
Godfrey Harold Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...
proved that \zeta\left(\tfrac+it\right) has infinitely many real zeros. The next two conjectures of Hardy and
John Edensor Littlewood John Edensor Littlewood (9 June 1885 – 6 September 1977) was a British mathematician. He worked on topics relating to analysis, number theory, and differential equations, and had lengthy collaborations with G. H. Hardy, Srinivasa Ramanu ...
on the distance between real zeros of \zeta\left(\tfrac+it\right) and on the density of zeros of \zeta\left(\tfrac+it\right) on the interval (T,T+H] for sufficiently large T > 0, and H = T^ and with as small as possible value of a > 0, where \varepsilon > 0 is an arbitrarily small number, open two new directions in the investigation of the Riemann zeta function: # For any \varepsilon > 0 there exists a lower bound T_0 = T_0(\varepsilon) > 0 such that for T \geq T_0 and H=T^ the interval (T,T+H] contains a zero of odd order of the function \zeta\bigl(\tfrac+it\bigr). Let N(T) be the total number of real zeros, and N_0(T) be the total number of zeros of odd order of the function ~\zeta\left(\tfrac+it\right)~ lying on the interval (0,T]~. #
  • For any \varepsilon > 0 there exists T_0 = T_0(\varepsilon) > 0 and some c = c(\varepsilon) > 0, such that for T \geq T_0 and H=T^ the inequality N_0(T+H)-N_0(T) \geq c H is true.

  • Selberg's zeta function conjecture

    investigated the problem of Hardy–Littlewood ''2'' and proved that for any ε > 0 there exists such T_0 = T_0(\varepsilon) > 0 and ''c'' = ''c''(ε) > 0, such that for T \geq T_0 and H=T^ the inequality N(T+H)-N(T) \geq cH\log T is true. Selberg conjectured that this could be tightened to H=T^. proved that for a fixed ε satisfying the condition 0 < ε < 0.001, a sufficiently large ''T'' and H = T^, a = \tfrac = \tfrac -\tfrac, the interval (''T'', ''T''+''H'') contains at least ''cH'' log(''T'') real zeros of the
    Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
    \zeta\left(\tfrac+it\right) and therefore confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba can not be improved in respect of the order of growth as ''T'' → ∞. proved that an analog of the Selberg conjecture holds for almost all intervals (''T'', ''T''+''H''], H = T^, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits to investigate zeros of the Riemann zeta function on "supershort" intervals of the critical line, that is, on the intervals (''T'', ''T''+''H''], the length ''H'' of which grows slower than any, even arbitrarily small degree ''T''. In particular, he proved that for any given numbers ε, \varepsilon_1 satisfying the conditions 0<\varepsilon, \varepsilon_<1 almost all intervals (''T'', ''T''+''H''] for H\ge\exp contain at least H(\log T)^ zeros of the function \zeta\left(\tfrac+it\right). This estimate is quite close to the one that follows from the Riemann hypothesis.


    Numerical calculations

    The function :\pi^\Gamma(\tfrac)\zeta(s) has the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functional equation, so one can prove the existence of zeros exactly on the real line between two points by checking numerically that the function has opposite signs at these points. Usually one writes :\zeta(\tfrac +it) = Z(t)e^ where Hardy's
    Z function In mathematics, the Z function is a function (mathematics), function used for studying the Riemann zeta function along the Riemann hypothesis, critical line where the argument is one-half. It is also called the Riemann–Siegel Z function, the R ...
    and the
    Riemann–Siegel theta function In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as :\theta(t) = \arg \left( \Gamma\left(\frac+\frac\right) \right) - \frac t for real values of ''t''. Here the argument is chosen in such a way th ...
    θ are uniquely defined by this and the condition that they are smooth real functions with θ(0)=0. By finding many intervals where the function ''Z'' changes sign one can show that there are many zeros on the critical line. To verify the Riemann hypothesis up to a given
    imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
    ''T'' of the zeros, one also has to check that there are no further zeros off the line in this region. This can be done by calculating the total number of zeros in the region using
    Turing's method In mathematics, Turing's method is used to verify that for any given Gram point there lie ''m'' + 1 zeros of , in the region , where is the Riemann zeta function. It was discovered by Alan Turing Alan Mathison Turing (; 23 Jun ...
    and checking that it is the same as the number of zeros found on the line. This allows one to verify the Riemann hypothesis computationally up to any desired value of ''T'' (provided all the zeros of the zeta function in this region are simple and on the critical line). Some calculations of zeros of the zeta function are listed below, where the "height" of a zero is the magnitude of its imaginary part, and the height of the ''n''th zero is denoted by γ''n''. So far all zeros that have been checked are on the critical line and are simple. (A multiple zero would cause problems for the zero finding algorithms, which depend on finding sign changes between zeros.) For tables of the zeros, see or .


    Gram points

    A
    Gram point The gram (originally gramme; SI unit symbol g) is a unit of mass in the International System of Units (SI) equal to one one thousandth of a kilogram. Originally defined as of 1795 as "the absolute weight of a volume of pure water equal to th ...
    is a point on the critical line 1/2 + ''it'' where the zeta function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + ''it'') = ''Z''(''t'')e − ''i''θ(''t''), where Hardy's function, ''Z'', is real for real ''t'', and θ is the
    Riemann–Siegel theta function In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as :\theta(t) = \arg \left( \Gamma\left(\frac+\frac\right) \right) - \frac t for real values of ''t''. Here the argument is chosen in such a way th ...
    , we see that zeta is real when sin(θ(''t'')) = 0. This implies that θ(''t'') is an integer multiple of π, which allows for the location of Gram points to be calculated fairly easily by inverting the formula for θ. They are usually numbered as ''g''''n'' for ''n'' = 0, 1, ..., where ''g''''n'' is the unique solution of θ(''t'') = ''n''π. Gram observed that there was often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observation Gram's law. There are several other closely related statements that are also sometimes called Gram's law: for example, (−1)''n''''Z''(''g''''n'') is usually positive, or ''Z''(''t'') usually has opposite sign at consecutive Gram points. The imaginary parts γ''n'' of the first few zeros (in blue) and the first few Gram points ''g''''n'' are given in the following table The first failure of Gram's law occurs at the 127th zero and the Gram point ''g''126, which are in the "wrong" order. A Gram point ''t'' is called good if the zeta function is positive at 1/2 + ''it''. The indices of the "bad" Gram points where ''Z'' has the "wrong" sign are 126, 134, 195, 211, ... . A ''Gram block'' is an interval bounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's law called Rosser's rule due to says that Gram blocks often have the expected number of zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals in the block may not have exactly one zero in them. For example, the interval bounded by ''g''125 and ''g''127 is a Gram block containing a unique bad Gram point ''g''126, and contains the expected number 2 of zeros although neither of its two Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in the first 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function. Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions. The distance of a zero from its expected position is controlled by the function ''S'' defined above, which grows extremely slowly: its average value is of the order of (log log ''T'')1/2, which only reaches 2 for T around 1024. This means that both rules hold most of the time for small ''T'' but eventually break down often. Indeed, showed that both Gram's law and Rosser's rule fail in a positive proportion of cases. To be specific, it is expected that in about 73% one zero is enclosed by two successive Gram points, but in 14% no zero and in 13% two zeros are in such a Gram-interval on the long run.


    Arguments for and against the Riemann hypothesis

    Mathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authors who express an opinion, most of them, such as and , imply that they expect (or at least hope) that it is true. The few authors who express serious doubt about it include , who lists some reasons for skepticism, and , who flatly states that he believes it false, that there is no evidence for it and no imaginable reason it would be true. The consensus of the survey articles (, , and ) is that the evidence for it is strong but not overwhelming, so that while it is probably true there is reasonable doubt. Some of the arguments for and against the Riemann hypothesis are listed by , , and , and include the following: * Several analogues of the Riemann hypothesis have already been proved. The proof of the Riemann hypothesis for varieties over finite fields by is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions associated with automorphic forms satisfy a Riemann hypothesis, which includes the classical Riemann hypothesis as a special case. Similarly
    Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
    s satisfy the analogue of the Riemann hypothesis, and are in some ways similar to the Riemann zeta function, having a functional equation and an infinite product expansion analogous to the Euler product expansion. But there are also some major differences; for example, they are not given by Dirichlet series. The Riemann hypothesis for the
    Goss zeta function In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. proved that it satisfies an analogue of the Riemann hypothesis. proved results for a higher-dimensiona ...
    was proved by . In contrast to these positive examples, some
    Epstein zeta function In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are ma ...
    s do not satisfy the Riemann hypothesis even though they have an infinite number of zeros on the critical line. These functions are quite similar to the Riemann zeta function, and have a Dirichlet series expansion and a
    functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
    , but the ones known to fail the Riemann hypothesis do not have an
    Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eu ...
    and are not directly related to
    automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
    s. * At first, the numerical verification that many zeros lie on the line seems strong evidence for it. But analytic number theory has had many conjectures supported by substantial numerical evidence that turned out to be false. See Skewes number for a notorious example, where the first exception to a plausible conjecture related to the Riemann hypothesis probably occurs around 10316; a counterexample to the Riemann hypothesis with imaginary part this size would be far beyond anything that can currently be computed using a direct approach. The problem is that the behavior is often influenced by very slowly increasing functions such as log log ''T'', that tend to infinity, but do so so slowly that this cannot be detected by computation. Such functions occur in the theory of the zeta function controlling the behavior of its zeros; for example the function ''S''(''T'') above has average size around (log log ''T'')1/2. As ''S''(''T'') jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expect any counterexamples to the Riemann hypothesis to start appearing only when ''S''(''T'') becomes large. It is never much more than 3 as far as it has been calculated, but is known to be unbounded, suggesting that calculations may not have yet reached the region of typical behavior of the zeta function. * Denjoy's probabilistic argument for the Riemann hypothesis is based on the observation that if μ(''x'') is a random sequence of "1"s and "−1"s then, for every , the
    partial sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
    s M(x) = \sum_ \mu(n) (the values of which are positions in a
    simple random walk In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
    ) satisfy the bound M(x) = O(x^) with probability 1. The Riemann hypothesis is equivalent to this bound for the
    Möbius function The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most oft ...
     μ and the
    Mertens function In number theory, the Mertens function is defined for all positive integers ''n'' as : M(n) = \sum_^n \mu(k), where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive re ...
    ''M'' derived in the same way from it. In other words, the Riemann hypothesis is in some sense equivalent to saying that μ(''x'') behaves like a random sequence of coin tosses. When μ(''x'') is nonzero its sign gives the parity of the number of prime factors of ''x'', so informally the Riemann hypothesis says that the parity of the number of prime factors of an integer behaves randomly. Such probabilistic arguments in number theory often give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answer for some results, such as
    Maier's theorem In number theory, Maier's theorem is a theorem about the numbers of primes in short intervals for which Cramér's probabilistic model of primes gives a wrong answer. The theorem states that if π is the prime-counting function and λ is greater ...
    . * The calculations in show that the zeros of the zeta function behave very much like the eigenvalues of a random
    Hermitian matrix In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
    , suggesting that they are the eigenvalues of some self-adjoint operator, which would imply the Riemann hypothesis. All attempts to find such an operator have failed. * There are several theorems, such as
    Goldbach's weak conjecture In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that : Every odd number greater than 5 can be expressed as the sum of three primes. (A prime ma ...
    for sufficiently large odd numbers, that were first proved using the generalized Riemann hypothesis, and later shown to be true unconditionally. This could be considered as weak evidence for the generalized Riemann hypothesis, as several of its "predictions" are true. * Lehmer's phenomenon, where two zeros are sometimes very close, is sometimes given as a reason to disbelieve the Riemann hypothesis. But one would expect this to happen occasionally by chance even if the Riemann hypothesis is true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted by Montgomery's conjecture. *
    Patterson Patterson may refer to: People * Patterson (surname) Places ;Canada * Pattersons Corners, Ontario *Patterson Township, Ontario *Patterson, Calgary a neighbourhood in Calgary, Alberta. ;United States of America * Patterson, Arkansas *Patterson, C ...
    suggests that the most compelling reason for the Riemann hypothesis for most mathematicians is the hope that primes are distributed as regularly as possible.p. 75: "One should probably add to this list the 'Platonic' reason that one expects the natural numbers to be the most perfect idea conceivable, and that this is only compatible with the primes being distributed in the most regular fashion possible..."


    Notes


    References

    There are several nontechnical books on the Riemann hypothesis, such as , , , , and . The books , , , and give mathematical introductions, while , and are advanced
    monograph A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject. In library cataloging, ''monograph ...
    s. * * * * * Reprinted in . * * * * * * * * * * * Reprinted in . * * * * * * * * * * * * * * Reprinted in . * Reprinted in . * * Reprinted in . *
    Review
    * * . Reprinted 1990, , * * (Reprinted by Dover 2003) * * * * * * * * * * * * * * *. * * * * * * Reprinted in . * * . * . * * * * * This unpublished book describes the implementation of the algorithm and discusses the results in detail. * * * * * * * . In ''Gesammelte Werke'', Teubner, Leipzig (1892), Reprinted by Dover, New York (1953)
    Original manuscript
    (with English translation). Reprinted in and * * * * ; see als
    announcement on Tao's blog
    January 19, 2018 * * * * Reprinted in . * * * * * * * * Reprinted in Gesammelte Abhandlungen, Vol. 1. Berlin: Springer-Verlag, 1966. * * * * * * * * * * Reprinted in . * * * Reprinted in . * * Reprinted in Oeuvres Scientifiques/Collected Papers by Andre Weil * * * *


    Popular expositions

    * * * * * * * Frenkel, Edward (2014)
    The Riemann Hypothesis
    Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
    , Mar 11, 2014 (video)


    External links

    * *
    American institute of mathematics The American Institute of Mathematics (AIM) is one of eight mathematical institutes in the United States, funded by the National Science Foundation (NSF). It was founded in 1994 by John Fry, co-founder of Fry's Electronics, and originally located ...

    Riemann hypothesis
    * Zeroe
    database
    103 800 788 359 zeroes
    ''The Key to the Riemann Hypothesis'' - Numberphile
    a
    YouTube YouTube is a global online video platform, online video sharing and social media, social media platform headquartered in San Bruno, California. It was launched on February 14, 2005, by Steve Chen, Chad Hurley, and Jawed Karim. It is owned by ...
    video about the Riemann hypothesis by
    Numberphile ''Numberphile'' is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its s ...
    * Poem about the Riemann hypothesis
    sung
    by
    John Derbyshire John Derbyshire (born 3 June 1945) is a British-born American far-right political commentator, writer, journalist and computer programmer. He was once known as a paleoconservative, until he was fired from the ''National Review'' in 2012 for wr ...
    . * (Slides for a lecture) * * * (Reviews the GUE hypothesis, provides an extensive bibliography as well). * includin
    papers on the zeros of the zeta function
    an

    * Slides of a talk * . A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros * * . * * * , a simple animated Java applet. * *
    Zetagrid
    ' (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005 {{Authority control 1859 introductions Analytic number theory Bernhard Riemann Conjectures Hilbert's problems Hypotheses Millennium Prize Problems Unsolved problems in number theory Zeta and L-functions