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 1 ...
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 for ...
s with
real part . Many consider it to be the most important
unsolved problem
List of unsolved problems may refer to several notable conjectures or open problems in various academic fields:
Natural sciences, engineering and medicine
* Unsolved problems in astronomy
* Unsolved problems in biology
* Unsolved problems in chem ...
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 integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
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, make up
Hilbert's eighth problem
Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns number theory, and in particular the Riemann hypothesis, although it is also concerned with the Goldbach Conjecture. The problem as st ...
in
David Hilbert's list of
twenty-three unsolved problems; it is also one of the
Clay Mathematics Institute'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 t ...
, 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.
The Riemann zeta function ζ(''s'') is a
function whose
argument ''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 and ''i'' is the
imaginary unit.
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 infinite series
:
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. He also proved that it equals the
Euler product
:
where the
infinite product 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
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 + \c ...
satisfy the relation
:
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
, 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 ), 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 of a complex-valued function of a complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if i ...
at ''s'' = 1.
In the strip this extension of the zeta function satisfies the
functional equation
:
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 th ...
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". 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,
:
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 of ...
. 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 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 function, periodic variation, typically in time, of some measure about a central value (often a point of Mechanical equilibrium, equilibrium) or between two or more different states. Familiar examples o ...
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 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 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 elem ...
of the real parts of the zeros, then
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
:
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 ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is ...
,
is the
logarithmic integral function, and
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
:
where
is
Chebyshev's second function.
proved that the Riemann hypothesis implies that for all
there is a prime
satisfying
:
.
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 functions, 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 of ...
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
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 Ram ...
, 1912; see for instance: paragraph 14.25 in ). (For the meaning of these symbols, see
Big O notation.) The
determinant 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
:
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 ''n'' > 5040 if and only if the Riemann hypothesis is true, where γ is the
Euler–Mascheroni constant.
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 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'' > 1, where
is the ''n''th
harmonic number.
The Riemann hypothesis is also true if and only if the inequality
: