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

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      https://www.ias.ac.in/article/fulltext/pmsc/104/01/0167-0176

    • Keywords

       

      Generalised Dirichlet series; distribution of zeros; neighbourhood of the critical line

    • 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<1/2and let {an}be a sequence of complex numbers satisfying the inequality$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1} $$for N = 1,2,3,…,also for n = 1,2,3,…let αnbe real and ¦αn¦ ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function$$\sum\limits_{n = 1}^N {a_n \left( {n + \alpha _n } \right)^{ - s} } = \zeta \left( s \right) + \sum\limits_{n = 1}^\infty {\left( {a_n \left( {n + \alpha _n } \right)^{ - s} - n^{ - s} } \right)} $$in the rectangle (1/2-δ⩽σ⩽1/2+δ,Tt⩽2T) (where 0<δ<1/2)isC(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided TT0(θ,δ)a large positive constant.

      Theorem B.In the above theorem we can relax the condition on an to$$\left| {\sum\limits_{m = 1}^N {a_m - N} } \right| \leqslant \left( {\frac{1}{2} - \theta } \right)^{ - 1} N^0 $$ and ¦aN¦ ≤ (1/2-θ)-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,Tt⩽2T)is > C(θ,δ) Tlog T(log logT)-1.The upper bound for the number of zeros in σ⩾1/3+δ,Tt⩽2T) isO(T)provided$$\sum\limits_{n \leqslant x} {a_n } = x + O_s \left( {x^2 } \right)$$for every ε > 0.

    • Author Affiliations

       

      R Balasubramanian1 K Ramachandra1 2

      1. Institute of Mathematical Sciences, Tharamani, Madras - 600113, India
      2. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay - 400005, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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