Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum\_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that $$\sum\_{n=1}^{\infty} \frac{f(n)}{n} \neq 0?$$ I ask this question in attempt to understand how general of a statement it is that $L(1 , \chi) \neq 0$ where $\chi$ is a non-principal real character. 



Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
Not necessarily. The first counterexample might be
$q=14$ and $f(n)=1, -1, -1, -1, -1, 1, 0, -1, 1, 1, 1, 1, -1, 0$
for $n=1,2,3,\ldots,14$.


