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 nonprincipal real character.

1$\begingroup$ I think it has been established in various ways that linear combinations of nice $L$functions do not reliably have nonvanishing properties that the individuals might have, although they'd obviously have the same growth properties. (E.g., the BombieriHejhal paper about linear combinations of the two idealclass characters for $\sqrt{5}$.) That is, RHtype results are not at all stable under linear combinations, but Lindeloftype results, or subconvexity results, would be. So, on general considerations already, I'd be surprised if such a thing were true... $\endgroup$ – paul garrett Jan 25 '15 at 0:39
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 $n=1,2,3,\ldots,14$.

$\begingroup$ Very cool..would you mind elaborating a little bit on how you found this? $\endgroup$ – George Shakan Jan 24 '15 at 22:25

3$\begingroup$ Thanks. Basically I just used the routine lindep in gp. Once one has guessed such an identity numerically it's usually not hard to prove. Here $f(n)$ is odd and 7antiperiodic, so $\sum_{n=1}^\infty f(n)/n = 0$ comes down to a linear relation among the cosecants of multiples of $\pi/7$, namely $$ \frac1{\sin \pi/7} = \frac1{\sin 2\pi/7} + \frac1{\sin 3\pi/7}. $$ $\endgroup$ – Noam D. Elkies Jan 25 '15 at 6:06
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.