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 ∑i=0q−1f(i) = 0. It is known that in this case the function κf(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 Λf(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 δf naturally appears in the representation of the function κf(s). In this article we find some asymptotic properties of the function κf(s), prove that κf(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 κf(s) has only trivial zeros which are simple.