On Dirichlet series with periodic coefficients (Q1850936)

The author investigates the distribution of zeros of the Dirichlet series \(L(s,f)=\sum_{n=1}^\infty\frac{f(n)}{n^s}\) with \(q\)-periodic coefficients \(f(n)\), i.e. \(f(n+q)=f(n)\) for all integers \(n\) and some fixed integer~\(q\). He finds the zero free regions and, similarly as for the Riemann zeta-function or the Lerch zeta-function, defines \textit{trivial} and \textit{nontrivial} zeros of \(L(s,f)\). Then he proves an asymptotic formula for the number of nontrivial zeros. Also he finds an asymptotic formula for the sum \(\sum_{| \gamma| \leq T}(\beta-1/2)\), where \(\beta+i\gamma\) denotes the nontrivial zero of \(L(s,f)\). This formula shows that some functions \(L(s,f)\) have asymmetric distribution of nontrivial zeros and thus they have nontrivial zeros with respect to the critical line. The example of such function is \(L(s,c_q)\), where \(c_q\) is the Ramanujan sum \(c_q(n)=\sum_{a\mod q, (a,q)=1}\exp(2\pi ian/q)\) . From the other side the author shows that nontrivial zeros of \(L(s,f)\) are clustered around the critical line.