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    • DirichletL-function and power series for Hurwitz zeta function

      V V Rane

      Volume 103 Issue 1 April 1993 pp 27-39

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/103/01/0027-0039

    • Keywords

       

      Hurwitz zeta function; DirichletL-function; power series

    • Abstract

       

      For 0 < α < 1, letζ(s, α) be the Hurwitz zeta function and let ζ1 (s, α) = ζ(s, α) -α− s. For a fixeds, we developζ1(s,α) as a power series in α in the complex circle ¦α¦ < 1. If$$\sum\limits_{\chi \left( {\bmod q} \right)} {L\left( {s,\chi } \right)L\left( {s',\bar \chi } \right)} = \frac{{\phi \left( q \right)}}{{q^{s + s'} }}\sum\limits_{k/q} \mu \left( {\frac{q}{k}} \right)\left( {\sum\limits_{a = 1}^k {\left( {\frac{k}{a}} \right)^{\operatorname{Re} s + \operatorname{Re} s'} + Q\left( {s,s',k} \right)} } \right)$$, we obtain an asymptotic expansion for Q(k) =Q(s,s′,k) using the power series forζ1(s,α)

    • Author Affiliations

       

      V V Rane1

      1. Department of Mathematics, The Institute of Science, 15, Madam Cama Road, Bombay - 400032, India

    • Dates

       
      Published
      April 1993
      Manuscript received
      5 November 1992
      Manuscript revised
      3 January 1993

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