Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-vanishing of Dirichlet series with periodic coefficients   For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we
study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known
that this admits an analytic continuation to the entire complex plane except at
$s=1$, where it has a simple pole with residue $$\rho:= q^{-1}\sum_{1\leq a\leq
q} f(a).$$ Thus, the function is analytic at $s=1$ when $\rho=0$ and in this
case, we study its non-vanishing using the theory of linear forms in logarithms
and Dirichlet $L$-series. In this way, we give new proofs of an old criterion
of Okada for the non-vanishing of $L(1,f)$ as well as a classical theorem of
Baker, Birch and Wirsing. We also give some new necessary and sufficient
conditions for the non-vanishing of $L(1,f)$.

27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph