TITLE: Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.
QUESTION [12 upvotes]: 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.

REPLY [16 votes]: On the other hand, a variant of the question has a positive answer.
This question was raised by Chowla in 1964 in the case that $q = p$ is prime and $f(p) = 0$ (but with $f$ taking arbitrary rational values). This was settled affirmatively in a paper by A. Baker, B. Birch, and E. Wirsing ("On a problem of Chowla," Journal of Number Theory, 1973), using Baker's theory of logarithmic linear forms. For a general modulus $q$ they prove:
If $f : \mathbb{N} \to \mathbb{Q}$ has a period $q$, satisfies $f(r) = 0$ for $(r,q) > 1$, and is not identically zero, then $\sum_{n} f(n)/n \neq 0$.
The condition $f(r) = 0$ for $(r,q) > 1$ is only natural as it is met by a primitive mod $q$ character.