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

Balasubramanian, R. ; Ramachandra, K. (1994) On the zeros of a class of generalised Dirichlet series-XIV Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 104 (1). pp. 167-176. ISSN 0253-4142  Preview
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Related URL: http://dx.doi.org/10.1007/BF02830880

## 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 {an} be a sequence of complex numbers satisfying the inequality |∑Nm=1am − N| ≤ (1/2 − θ)−1 for N = 1,2,3,...,also for n = 1,2,3,...,let α n be real and |αn| ≤ C(θ) where C(θ) > 0 is a certain (small)constant depending only on θ. Then the number of zeros of the function ∑Nn=1an (n + αn)−s = ζ (s) + ∑n=1 (an(n + αn)−s − n−s) in the rectangle (½−δ ≤ δ ≤ ½+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 ≥ T0(θ,δ) a large positive constant. Theorem B. In the above theorem we can relax the condition on a n to |∑Nm=1am − N| ≤ (½ −θ)−1 Nφ and |aN| ≤ (½−θ)−1. Then the lower bound for the number of zeros in (σ ≥ ½−δ,T ≤ t ≤ 2T) is > C(θ,δ) Tlog T(log logT)−1. The upper bound for the number of zeros in σ ≥ ½ +δ,T ≤ t ≤ 2T) is O(T) provided ∑n≤xan = x + Os(x2) for every ε > 0.

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