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.
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).
(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.)
The generalized Riemann hypothesis (for Dirichlet L-functions) was probably formulated for the first time by Adolf Piltz in 1884.
[1] Like the original Riemann hypothesis, it has far reaching consequences about the distribution of prime numbers.
The generalized Riemann hypothesis asserts that, for every Dirichlet character χ and every complex number s with L(χ, s) = 0, if s is not a negative real number, then the real part of s is 1/2.
This is often used in proofs, and it has many consequences, for example (assuming GRH): If GRH is true, then for every prime p there exists a primitive root mod p (a generator of the multiplicative group of integers modulo p) that is less than
[5] Assuming the truth of the GRH, the estimate of the character sum in the Pólya–Vinogradov inequality can be improved to
The Dedekind zeta-function of K is then defined by for every complex number s with real part > 1.
The Dedekind zeta-function satisfies a functional equation and can be extended by analytic continuation to the whole complex plane.
The ordinary Riemann hypothesis follows from the extended one if one takes the number field to be Q, with ring of integers Z.
The ERH implies an effective version[6] of the Chebotarev density theorem: if L/K is a finite Galois extension with Galois group G, and C a union of conjugacy classes of G, the number of unramified primes of K of norm below x with Frobenius conjugacy class in C is where the constant implied in the big-O notation is absolute, n is the degree of L over Q, and Δ its discriminant.