
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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
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 and its analytic c ...
has its
zeros only at the negative
even integer
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers.
The ...
s 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 for ...
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 is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
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 list of unsolved problems in mathematics, unsolved problems in number theory and all of mathematics. It states that every even and odd numbers, even natural number greater than 2 is the ...
and the
twin prime conjecture, make up
Hilbert's eighth problem in
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
's list of
twenty-three unsolved problems; it is also one of the
Millennium Prize Problems
The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...
of the
Clay Mathematics Institute, which offers
US$
The United States dollar (Currency symbol, symbol: Dollar sign, $; ISO 4217, currency code: USD) is the official currency of the United States and International use of the U.S. dollar, several other countries. The Coinage Act of 1792 introdu ...
1 million for a solution to any of them. The name is also used for some closely related analogues, such as the
Riemann hypothesis for curves over finite fields.
The Riemann zeta function ''ζ''(''s'') is a
function whose
argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
''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, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
and ''i'' is the
imaginary unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
.
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 and its analytic c ...
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 said ...
infinite series
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
:
Leonhard Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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 E ...
:
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 the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound ...
extends over all prime numbers ''p''.
The Riemann hypothesis discusses zeros outside the
region of convergence 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, all choices of how to perform this analytic continuation will lead to the same result, by the
identity theorem. A first step in this continuation observes that the series for the zeta function and the
Dirichlet eta function satisfy the relation
:
within the region of convergence for both series. But the eta 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
, extending it from to a larger domain: , except for the points where
is zero. These are the points
where
can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see
Dirichlet eta function#Landau's problem with ζ(s) = η(s)/0 and solutions), giving a finite value for all values of ''s'' with positive real part except the
simple pole 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 ...
:
One may then define ''ζ''(''s'') for all remaining nonzero complex numbers ''s'' ( and ) by applying this equation outside the strip, and letting ''ζ''(''s'') equal the right side of the equation whenever ''s'' has non-positive real part (and ).
If ''s'' is a negative even integer, then , because the factor sin(''s''/2) vanishes; these are the zeta function's ''trivial zeros''. (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 opposite th ...
function are canceled by the poles of the
gamma function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
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 nontrivial zeros lie in the ''critical strip'' where ''s'' has real part between 0 and 1.
File:ParametricZeta.svg, Riemann zeta function along the critical line with . Real values are shown on the horizontal axis and imaginary values are on the vertical axis. , is plotted with ''t'' ranging between −30 and 30.
File:Riemann3d Re 0.1 to 0.9 Im 1 to 51.ogg, Animation showing in 3D the Riemann zeta function critical strip (blue, where ''s'' has real part between 0 and 1), critical line (red, for real part of ''s'' equals 0.5) and zeroes (cross between red and orange): 'x'',''y'',''z''= e(''ζ''(''r'' + ''it'')), Im(''ζ''(''r'' + ''it'')), ''t''with and
File:RiemannCriticalLine.svg, The real part (red) and imaginary part (blue) of the Riemann zeta function ''ζ''(''s'') along the critical line in the complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
with real part . The first nontrivial
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or a particularly simple object possessing a given structure (e.g., group (mathematics), group, topological space). The n ...
zeros, where ''ζ''(''s'') equals zero, occur where both curves touch the horizontal x-axis, for complex numbers with imaginary parts Im(''s'') equaling ±14.135, ±21.022 and ±25.011.
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 a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte ...
". His formula was given in terms of the related function
:
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 genera ...
,
:
where ''μ'' is the
Möbius function
The Möbius function \mu(n) 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 m ...
. Riemann's formula is then
:
where the sum is over the nontrivial zeros of the zeta function and where Π
0 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:
:
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 theory, number theoretic significance. In particular, according to the prime number the ...
given by the
Cauchy principal value of the divergent integral
:
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 for
the number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, 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 analysis, 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 p ...
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 every element of .
Dually, a lower bound or minorant of is defined to be an element of that is less ...
of the real parts of the zeros, then
, where
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 . It is denoted by (unrelated to the number ).
A symmetric variant seen sometimes is , which is equal ...
,
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 theory, number theoretic significance. In particular, according to the prime number the ...
,
is the
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
of ''x'', and
big O notation
Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
is used here.
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 von Koch's result, due to , says that the Riemann hypothesis implies
:
also showed that the Riemann hypothesis implies
:
where
is
Chebyshev's second function.
proved that the Riemann hypothesis implies that for all
there is a prime
satisfying
:
.
The constant 4/ may
be reduced to (1 + ''ε'') provided that ''x'' is taken to be sufficiently large.
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 generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...
s, in addition to the primes counting function above.
One example involves the
Möbius function
The Möbius function \mu(n) 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 m ...
''μ''. The statement that the equation
:
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 is defined by
:
then the claim that
:
for every positive ''ε'' is equivalent to the Riemann hypothesis (
J. E. Littlewood, 1912; see for instance: paragraph 14.25 in ). The
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
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. Littlewood's result has been improved several times since then, by
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 Leopo ...
,
Edward Charles Titchmarsh
Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading British mathematician.
Education
Titchmarsh was educated at King Edward VII School (Sheffield) and Balliol College, Oxford, where he began his studies in October 1 ...
, Helmut Maier and
Hugh Montgomery, and
Kannan Soundararajan
Kannan Soundararajan (born December 27, 1973) is an Indian-born American mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan, where he had also ...
. Soundararajan's result is that, conditional on the Riemann hypothesis,
:
The Riemann hypothesis puts a rather tight bound on the growth of ''M'', since disproved the slightly stronger
Mertens conjecture
:
Another closely related result is due to , that the Riemann hypothesis is equivalent to the statement that the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...
of the
simplicial complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
determined by the lattice of integers under divisibility is
for all
(see
incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set
and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
).
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, which states that if ''σ''(''n'') is the
sigma function, given by
:
then
:
for all if and only if the Riemann hypothesis is true, where ''γ'' is the
Euler–Mascheroni constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarith ...
.
A related bound was given by
Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:
:
for every
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
, where
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
: