Zeros of Dirichlet series with periodic coefficients

Let a = (an)n≥1 be a periodic sequence, Fa(s) the meromorphic continuation of P n≥1 an/n , and Na(σ1, σ2, T ) the number of zeros of Fa(s), counted with their multiplicities, in the rectangle σ1 < Re s < σ2, | Im s| ≤ T . We extend previous results of Laurinčikas, Kaczorowski, Kulas, and Steuding, by showing that if Fa(s) is not of the form P (s)Lχ(s), where P (s) is a Dirichlet polynomial and Lχ(s) a Dirichlet L-function, then there exists an η = η(a) > 0 such that for all 1/2 < σ1 < σ2 < 1+ η, we have c1T ≤ Na(σ1, σ2, T ) ≤ c2T for sufficiently large T , and suitable positive constants c1 and c2 depending on a, σ1, and σ2.