Generalized Riemann hypothesis

The generalized Riemann hypothesis conjectures that nontrivial zeros of a broad family of L-functions lie on the line with real part one-half.

Riemann zeta function: The complex function defined initially by the Dirichlet series ∑ n⁻ˢ and continued meromorphically to the complex plane. Its zero conjecture is the original case that generalized the hypothesis extends.
Completed L-function: An L-function multiplied by specified gamma and conductor factors so that it satisfies a symmetric functional equation. Completion exposes the symmetry that pairs zeros around the critical line.
Prime number theorem for arithmetic progressions: The theorem that primes are asymptotically equidistributed among reduced residue classes modulo a fixed integer. Dirichlet L-function zeros govern error terms in this distribution.
Riemann hypothesis: The conjecture that every nontrivial zero of the Riemann zeta function has real part one-half. It is the single-function case at the center of the generalized statement.
Riemann hypothesis verification: Computational and theoretical checks of the Riemann hypothesis for specified ranges or collections of zeros. Finite verification supports evidence but cannot establish an infinite zero-location claim.
Dirichlet L-function: A complex function formed from a Dirichlet character and a Dirichlet series, with analytic continuation and a functional equation. These functions supply the standard number-theoretic cases of the generalized conjecture.
Euler product: A product representation of certain L-functions over primes, encoding multiplicative arithmetic in an analytic function. It links L-function behavior to primes where the defining product converges.
Chebotarev density theorem: A theorem describing the densities of prime ideals with specified Frobenius conjugacy classes in a Galois extension. Artin and Dedekind L-functions connect its prime distribution estimates to generalized zero hypotheses.
Extended Riemann hypothesis: A conjecture asserting the Riemann-hypothesis zero condition for Dedekind zeta functions of number fields. It names a narrower, prominent subfamily of the generalized conjecture.
Random matrix theory: The study of statistical properties of matrices with randomly chosen entries or structure. Random matrix models predict statistical patterns for zeros of several L-function families.

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