Balasubramanian, R. ; Ramachandra, K. (1994) On the zeros of a class of generalised Dirichlet seriesXIV Proceedings of the Indian Academy of Sciences  Mathematical Sciences, 104 (1). pp. 167176. ISSN 02534142

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Official URL: http://www.ias.ac.in/j_archive/mathsci/104/1/1671...
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 {a_{n}} be a sequence of complex numbers satisfying the inequality ∑^{N}_{m=1}a_{m} − N ≤ (1/2 − θ)^{−1} for N = 1,2,3,...,also for n = 1,2,3,...,let α _{n} be real and α_{n} ≤ C(θ) where C(θ) > 0 is a certain (small)constant depending only on θ. Then the number of zeros of the function ∑^{N}_{n=1}a_{n} (n + α_{n})^{−s} = ζ (s) + ∑^{∞}_{n=1} (a_{n}(n + α_{n})^{−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 ≥ T_{0}(θ,δ) a large positive constant. Theorem B. In the above theorem we can relax the condition on a _{n} to ∑^{N}_{m=1}a_{m} − N ≤ (½ −θ)^{−1} N^{φ} and a_{N} ≤ (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≤x}a_{n} = x + O_{s}(x^{2}) for every ε > 0.
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Source:  Copyright of this article belongs to Indian Academy of Sciences. 
Keywords:  Generalised Dirichlet Series; Distribution of Zeros; Neighbourhood of the Critical Line 
ID Code:  1278 
Deposited On:  04 Oct 2010 07:56 
Last Modified:  16 May 2016 12:25 
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