The Riemann Hypothesis for Dirichlet $L$ Functions

TL;DR

This paper explores the relationship between the zeros of the Riemann zeta function and the Dirichlet beta function, establishing that the Riemann hypothesis for the beta function is equivalent to that for the zeta function.

Contribution

It demonstrates the equivalence of the Riemann hypothesis for the Dirichlet beta function and the Riemann zeta function, linking their zero distributions.

Findings

Riemann hypothesis holds for the Dirichlet beta function if and only if it holds for the Riemann zeta function.

Establishes a direct connection between zeros of two important Dirichlet L functions.

Provides a new perspective on the Generalized Riemann Hypothesis.

Abstract

This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $\zeta (s)$- a particular case of the Generalized Riemann Hypothesis.