# On the non-vanishing of certain Dirichlet series

## TL;DR

This paper investigates conditions under which certain Dirichlet series do not vanish at s=1, revealing cases where non-trivial odd rational-valued functions lead to zero values and implications for the linear independence of specific L-values.

## Contribution

It characterizes when the Dirichlet series associated with periodic functions vanish at s=1, providing new insights into the structure of these series and their relation to character L-values.

## Key findings

- No odd rational-valued functions lead to zero when (k,p-1)=1 or (k,p-1)=2 with p≡3 mod 4.
- Existence of odd functions with zero Dirichlet series in other cases.
- Values of L(1,χ)^2 for odd characters are linearly independent over Q.

## Abstract

Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and $p\equiv 3\mod 4$, then there are no odd rational-valued functions $f\not\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases there are examples of odd functions $f$ such that $D_k(1,f)=0$. As a consequence, we obtain, for example, that the set of values $L(1,\chi)^2$, where $\chi$ ranges over odd characters mod $p$, are linearly independent over $\mathbb Q$.