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.
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1$\begingroup$ I think it has been established in various ways that linear combinations of nice $L$-functions do not reliably have non-vanishing properties that the individuals might have, although they'd obviously have the same growth properties. (E.g., the Bombieri-Hejhal paper about linear combinations of the two ideal-class characters for $\sqrt{-5}$.) That is, RH-type results are not at all stable under linear combinations, but Lindelof-type results, or subconvexity results, would be. So, on general considerations already, I'd be surprised if such a thing were true... $\endgroup$– paul garrettCommented Jan 25, 2015 at 0:39
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2 Answers
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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 $n=1,2,3,\ldots,14$.
answered Jan 24, 2015 at 20:45
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1$\begingroup$ Very cool..would you mind elaborating a little bit on how you found this? $\endgroup$ Commented Jan 24, 2015 at 22:25
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4$\begingroup$ 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 7-antiperiodic, so $\sum_{n=1}^\infty f(n)/n = 0$ comes down to a linear relation among the cosecants of multiples of $\pi/7$, namely $$ \frac1{\sin \pi/7} = \frac1{\sin 2\pi/7} + \frac1{\sin 3\pi/7}. $$ $\endgroup$ Commented Jan 25, 2015 at 6:06
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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.
answered Jan 24, 2015 at 20:51