Dirichlet Series Associated with Strongly q-Multiplicative Functions

Abstract

In this work the Dirichlet series $\kappa {\text{f(}}s{\text{) = }}\sum\nolimits_n^\infty {\frac{{f(n - 1)}}{{n^s }}}$ associated with real strongly q-multiplicative functions f(n) are studied. We will confine ourselves to the case ∑ ^q−1_i=0 f(i) = 0. It is known that in this case the function κ _f (s) has an analytic continuation to the whole complex plane as an entire function with trivial zeros on the negative real line. The real function Λ _f (t) satisfying the integral equation with delayed argument $\delta _f \Lambda _f (\frac{t}{q}) = \int_0^t {\Lambda _f (u) du}$ for some nonzero real δ _f naturally appears in the representation of the function κ _f (s). In this article we find some asymptotic properties of the function κ _f (s), prove that κ _f (s) is an entire function of order 2, and also prove that in the region $\Re s \leqslant - k_0 ,|\Im s| \leqslant \frac{\pi }{{2\ln q}}$ the function κ _f (s) has only trivial zeros which are simple.