Zeta-Function
Zeta-function is the name given to certain functions of the complex variable s = + it that play a fundamental role in analytic number theory. The most important example is the Riemann zeta-function z(s). In the right half plane {s ∈ C: 1 < } the Riemann zeta-function is defined by the infinite series
It can be shown that the infinite series defining the zeta-function converges absolutely and uniformly on all compact subsets of {s ∈ C: 1 < } and therefore the Riemann zeta-function is an analytic function in this domain. Series of this type are called Dirchlet series. More generally, if a(1), a(2), a(3), ... is a sequence of complex numbers then
is the associated Dirichlet series. The natural domain of convergence for such a series is always a right half plane, but the half plane may be empty or all of C. If the half plane is not empty then the series defines an analytic function ƒ(s) in the interior of the half plane of convergence. The most important examples of Dirichlet series that occur in analytic number theory are those in which the coefficients a(n) carry arithmetic information. For example, in the half plane {s ∈ C: 1 < } the square of the Riemann zeta-function is given by the Dirchlet series
where for each positive integer n the coefficient d(n) is the number of positive integers that divide n.
The Riemann zeta-function is important in analytic number theory because it has a second representation in the half plane {s ∈ C: 1 < } as a convergent infinite product over the sequence of prime numbers. More precisely, we have
at each complex number s with 1 < , where the product is over the sequence p = 2, 3, 5, 7, 11, ... of prime numbers. It can be shown that the partial products converge uniformly on compact subsets of the half plane {s ∈ C: 1 < } and converge to exactly the same value as the Dirichlet series for z(s). That is, we have
for all complex numbers s = + it with 1 < . The identity amounts to an analytic formulation of the fundamental theorem of arithmetic, which states that each integer n 2 has a unique representation as a product of prime numbers. Because of this identity methods from complex analysis can be used to investigate problems about the distribution of prime numbers. For example, if (x) is defined to be the number of primes less than or equal to the positive real number x, then the prime number theorem asserts that
The first complete proof of the prime number theorem was given in 1896 by Jacques Hadamard and, independently, by C. J. de la Vallèe Poussin. The crucial step in both of their proofs was the discovery that z(s) 0 in a certain open subset of C that includes the closed half plane {s ∈ C: 1 }.
It can be shown that the Riemann zeta-function extends by analytic continuation to a function that is analytic on the complex plane C except for a pole of order 1 and residue 1 at the point s = 1. If we continue to write z(s) for the extended function then z(s) satisfies the functional equation
for all complex s. Here G(s) is Euler's gamma function, defined by
and is Euler's constant, defined by
The function G(s/2) has poles at the nonpositive even integers. This fact together with the functional equation shows that z(s) has a zero at each negative even integer. It is also known that z(s) has infinitely many nonreal zeros = + i that satisfy 0 < < 1. (The notation = + i for a nonreal zero is generally used. Of course in this context does not refer to Euler's constant.) It was conjectured by Bernhard Riemann in 1859 that z(s) 0 in the open half plane {s ∈ C: ½ < }, but this has never been proved. The conjecture is known as the Riemann hypothesis. An equivalent form of the Riemann hypothesis is the assertion that all of the nonreal zeros of z(s) occur on the line = ½, that is, if = + i is a nonreal zero of the zeta-function then = ½. The Riemann hypothesis is one of the most important unsolved problems in mathematics.
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