• Fulltext

       

        Click here to view fulltext PDF


      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/106/03/0217-0226

    • Keywords

       

      Quadratic forms; zeta-function; zeros near the line sigma equal to half

    • Abstract

       

      LetQ(u1,…,u1) =Σdijuiuj (i,j = 1 tol) be a positive definite quadratic form inl(≥3) variables with integer coefficientsdij(=dji). Puts=σ+it and for σ>(l/2) write$$Z_Q (s) = \Sigma '(Q(u_1 ,...,u_l ))^{ - s} ,$$ where the accent indicates that the sum is over alll-tuples of integer (u1,…,ul) with the exception of (0,…, 0). It is well-known that this series converges for σ>(l/2) and that (s-(l/2))ZQ(s) can be continued to an entire function ofs. Let σ be any constant with 0<σ<1/100. Then it is proved thatZQ(s)has ≫δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).

    • Author Affiliations

       

      K Ramachandra1 2 A Sankaranarayanan1

      1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai - 400 005, India
      2. National Institute of Advanced Studies, IISc Campus, Bangalore - 560 012, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

    • Editorial Note on Continuous Article Publication

      Posted on July 25, 2019

      Click here for Editorial Note on CAP Mode

© 2021-2022 Indian Academy of Sciences, Bengaluru.