The Generalized Riemann Hypothesis from zeros of a single L-function

TL;DR

This paper introduces a new hypothesis related to zeros of individual L-functions, demonstrating its equivalence to the Generalized Riemann Hypothesis for each primitive Dirichlet character, thus providing a novel perspective on this longstanding conjecture.

Contribution

The paper formulates a hypothesis based on zeros of a single L-function and proves its equivalence to the GRH for all primitive Dirichlet characters, offering a new approach to the conjecture.

Findings

${ m GRH}^\u2020[\u03a7]$ is equivalent to GRH for each character

The hypothesis links zeros of individual L-functions to the broader GRH

Provides a new criterion for the validity of GRH

Abstract

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is equivalent to the Generalized Riemann Hypothesis.