Generalized Riemann Hypothesis - Generalized Riemann Hypothesis (GRH)

Generalized Riemann Hypothesis (GRH)

The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf 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

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 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.

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