Generalized Riemann hypothesis

In mathematics, the extended Riemann hypothesis or generalized Riemann hypothesis is a generalization of the Riemann hypothesis with far reaching consequences about the distribution of prime numbers. Like the Riemann hypothesis, it is a conjecture and has not yet been proved; most mathematicians working in the field believe it to be true. The conjecture was probably formulated for the first time by Piltz in 1884.

The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that every value χ(n) is either 0 or a root of unity and such that there exists a positive integer k with χ(n + k) = χ(n) for all n. If such a character is given, we define the corresponding L-series by

$L(\chi ,s)=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}$

for every complex number s with real part > 1. By analytic continuation, this function can be extended to a meromorphic function defined on the whole complex plane. The generalized Riemann hypothesis then states that every zero of L(χ,s) with 0 ≤ Re(s) ≤ 1 has real part 1/2.

The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.

Consequences

An arithmetic progression in the natural numbers is a set of numbers of the form a, a+d, a+2d, a+3d, ... where a and d are natural numbers and d is non-zero. Dirichlet's theorem states that if a and d are coprime, then such an arithmetric progression contains infinitely many prime numbers. Let π(x,a,d) denote the number of prime numbers in this progression which are less than or equal to x. If the extended Riemann hypothesis is true, then for every a and d and for every ε > 0

$\pi (x,a,d)={\frac {1}{\phi (d)}}\int _{2}^{x}{\frac {1}{\ln x}}\,dx+O(x^{1/2+\epsilon })\quad {\mbox{ as }}\ x\to \infty$

where φ(d) denotes Euler's phi function and O is the Landau symbol. This is a considerable strengthening of the prime number theorem.