Zeros of certain Dirichlet series (Q5902749)

Let \(\alpha =a/q\), \((a,q)=1\), and define \(Z_ 1(s,\alpha)=\sum^{\infty}_{m=1, (m,q)=1}e^{2\pi im\alpha}m^{-s}\), \(Z_ 2(s,\alpha)=\sum^{\infty}_{m=1}e^{2\pi im\alpha}m^{-s}\). These series can be written as linear combinations of Dirichlet L- functions, which gives their analytic continuations. By theorem 1, the number of zeros of \(Z_ 1(s,\alpha)\) in the region \((1/2<) \sigma_ 1<\sigma <\sigma_ 2 (<1)\), \(| t| <T\) is \(\geq c(\alpha,\sigma_ 1,\sigma_ 2) T\), and by theorem 2 this holds also for \(Z_ 2(s,\alpha)\) if there are at least two primitive characters (mod q). Further, by theorem 3, the number of zeros of \(Z_ 1(s,\alpha)\) in the region \(\sigma >1\), \(| t| <T\) is \(\geq c(\alpha) T\) if q is square-free, and by theorem 4 the same is true for \(Z_ 2(s,\alpha)\) if q is a prime. The proofs are based on the universality theorem of S. M. Voronin for L-functions.