Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$-functions

TL;DR

This paper constructs Dirichlet series with periodic coefficients that satisfy Riemann's functional equation and explores the distribution of their zeros, linking them to the Generalized Riemann Hypothesis for Dirichlet L-functions.

Contribution

It introduces a new class of Dirichlet series satisfying Riemann's functional equation and characterizes their zeros in relation to the GRH.

Findings

The constructed series satisfy Riemann's functional equation for even characters.

Zeros on the critical line correspond to the GRH for real characters.

Non-real characters lead to zeros off the critical line.

Abstract

In this paper, we give Dirichlet series with periodic coefficients that have Riemann's functional equation and real zeros of Dirichlet $L$-functions. The details are as follows. Let $L(s,\chi)$ be the Dirichlet $L$-function and $G(\chi)$ be the Gauss sum associate with a primitive Dirichlet character $\chi$ (${\rm{mod}} \,\, q$). Put $f (s,\chi) := q^s L(s,\chi) + i^{-\kappa (\chi)} G(\chi) L(s,\overline{\chi})$, where $\overline{\chi}$ is the complex conjugate of $\chi$ and $\kappa (\chi) :=(1-\chi (-1))/2$. Then we prove that $f (s,\chi)$ satisfies Riemann's functional equation appearing in Hamburger's theorem if $\chi$ is even. In addition, we show that $f (\sigma,\chi) \ne 0$ all $\sigma \ge 1$. Moreover, we prove that $f(\sigma,\chi) \ne 0$ for all $1/2 \le \sigma < 1$ if and only if $L(\sigma,\chi) \ne 0$ for all $1/2 \le \sigma < 1$. When $\chi$ is real, all zeros of $f(s,\chi)$ with $\Re (s) >0$ are on the line $\sigma =1/2$ if and only if GRH for $L(s,\chi)$ is true. However, $f (s,\chi)$ has infinitely many zeros off the critical line $\sigma =1/2$ if $\chi$ is non-real.