Zeros of Dirichlet series with periodic coefficients

TL;DR

This paper investigates the zeros of Dirichlet series with periodic coefficients, establishing bounds on their distribution in certain vertical strips, and extends previous results by characterizing when zeros are abundant.

Contribution

It extends prior work by showing that unless the series is a product of a Dirichlet polynomial and an L-function, the zeros are linearly numerous in specified regions.

Findings

Zeros are linearly bounded in vertical strips for non-trivial series.

If not of the form P(s) L_χ(s), the zero count grows proportionally with T.

Provides conditions under which zeros are densely distributed.

Abstract

Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(\sigma_1, \sigma_2, T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle $\sigma_1 < \Re s < \sigma_2$, $|\Im s | \le T$. We extend previous results of Laurin\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s) L_{\chi} (s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\chi}(s)$ a Dirichlet L-function, then there exists an $\eta=\eta(a)>0$ such that for all $1/2 < \sigma_1 < \sigma_2 < 1+\eta$, we have $c_1 T \le N_a(\sigma_1, \sigma_2, T) \le c_2 T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, $\sigma_1$, and $\sigma_2$.