Abstract
LetQ(u 1,…,u 1) =Σd ij u i u j (i,j = 1 tol) be a positive definite quadratic form inl(≥3) variables with integer coefficientsd ij (=d ji ). Puts=σ+it and for σ>(l/2) write
where the accent indicates that the sum is over alll-tuples of integer (u 1,…,u l ) with the exception of (0,…, 0). It is well-known that this series converges for σ>(l/2) and that (s-(l/2))Z Q (s) can be continued to an entire function ofs. Let σ be any constant with 0<σ<1/100. Then it is proved thatZ Q (s)has ≫δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).
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Ramachandra, K., Sankaranarayanan, A. Hardy’s theorem for zeta-functions of quadratic forms. Proc. Indian Acad. Sci. (Math. Sci.) 106, 217–226 (1996). https://doi.org/10.1007/BF02867431
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DOI: https://doi.org/10.1007/BF02867431