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, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these L-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the function field case (not the number field case).
Global L-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis (ERH) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis (GRH). These two statements will be discussed in more detail below. (Many mathematicians use the label generalized Riemann hypothesis to cover the extension of the Riemann hypothesis to all global L-functions, not just the special case of Dirichlet L-functions.)
Generalized Riemann hypothesis (GRH)
The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Piltz in 1884. Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.
The formal statement of the hypothesis follows. A Dirichlet character is a completely multiplicative arithmetic function χ such that there exists a positive integer k with χ(n + k) = χ(n) for all n and χ(n) = 0 whenever gcd(n, k) > 1. If such a character is given, we define the corresponding Dirichlet L-function by
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 asserts that for every Dirichlet character χ and every complex number s with L(χ,s) = 0: if the real part of s is between 0 and 1, then it is actually 1/2.
The case χ(n) = 1 for all n yields the ordinary Riemann hypothesis.
Consequences of GRH
Dirichlet's theorem states that if a and d are coprime natural numbers, then the arithmetic progression a, a+d, a+2d, a+3d, … 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 generalized Riemann hypothesis is true, then for every coprime a and d and for every ε > 0
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