On the non-vanishing of certain Dirichlet series

Abstract

Given $k\in \mathbb{N}$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum _{n\ge 1}{d}_k(n)f(n)n^{-s}$$ 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 $p\equiv 3\mathrm{mod}4$, 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 $\mathbb{Q}$.

16 pages