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

• # Fulltext

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&gt;(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&gt;(l/2) and that (s-(l/2))ZQ(s) can be continued to an entire function ofs. Let σ be any constant with 0&lt;σ&lt;1/100. Then it is proved thatZQ(s)has ≫δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).

• # Author Affiliations

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

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019