• A Sankaranarayanan

      Articles written in Proceedings – Mathematical Sciences

    • Hardy’s theorem for zeta-functions of quadratic forms

      K Ramachandra A Sankaranarayanan

      More Details Abstract Fulltext PDF

      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).

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