  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 $L(1,f)$.
