Detecting zeros of Dirichlet $L$-functions via the Riemann zeta-function

TL;DR

Under the assumptions of the Riemann hypothesis, the paper establishes a link between the zeros of the Riemann zeta-function and Dirichlet L-functions, enabling detection of zeros of the latter via the former.

Contribution

It demonstrates that a specific vertical distribution of zeta zeros is equivalent to the generalized Riemann hypothesis and provides a method to detect Dirichlet L-function zeros through zeta zeros.

Findings

Vertical zero distribution relates to the generalized Riemann hypothesis.

Zeros of Dirichlet L-functions can be identified via zeta zeros under hypotheses.

Equivalence between zero distribution and the generalized Riemann hypothesis.

Abstract

Assuming the Riemann hypothesis, we show that a certain vertical distribution of the nontrivial zeros of the Riemann zeta-function is equivalent to the generalized Riemann hypothesis for Dirichlet $L$-functions. Furthermore, under both the Riemann hypothesis and the generalized Riemann hypothesis, we show that the nontrivial zeros of Dirichlet $L$-functions can be detected through those of the Riemann zeta-function.