Proof of generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions

TL;DR

This paper presents a short proof of the generalized Riemann hypothesis for Dedekind zeta functions and Dirichlet L-functions, based on Hecke's functional equation and algebraic methods, implying the hypothesis for these functions.

Contribution

It provides a novel, concise proof of the generalized Riemann hypothesis for Dedekind zetas and Dirichlet L-functions using algebraic and functional equation techniques.

Findings

Proof confirms gRH for Dedekind zeta functions

gRH for Dirichlet L-functions follows from Dedekind zeta case

Method simplifies previous approaches to gRH

Abstract

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the Riemann hypothesis derived in [$M_{A}$] (algebraic proof of the Riemann hypothesis) is given. The generalized Riemann hypothesis for Dirichlet L-functions is an immediately consequence of (gRH) for $\zeta_{k}$ and suitable product formula which connects the Dedekind zetas with L-functions.