On the zeros of a class of generalised Dirichlet series- XIV

Abstract

We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples.Theorem A.Let 0<θ <1/2 and let {a n }be a sequence of complex numbers satisfying the inequality ∑^N_n=1aⁿ(n+aⁿ)^-s=ζ(s)+∑^∞_n=1(aⁿ(n+aⁿ)^-s-n^-s)in the rectangle ½-δ≤ σ≤1/2+δ,T≤t≤2T)(where 0 <δ<1/2)is ≥C(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided T ≥T 0(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a n to and |aN|≤ (1/2-θ)^-1.Then the lower bound for the number of zeros in (σ≥ ½+δ,T≤t≤2T is O(T)provided for every ε > 0.