Hardy's theorem for zeta-functions of quadratic forms

Abstract

Let Q(u₁,…,u₁)=Σd_iju_iu_j (i,j=1 to l) be a positive definite quadratic form in l(≥3) variables with integer coefficients d_ij (=d_ji). Put s=σ+it and for σ > (½) write Z_Q(s)=Σ'(Q(u₁,...,u_l))^−s where the accent indicates that the sum is over all l-tuples of integer (u₁,…,u_l) with the exception of (0,…,0). It is well-known that this series converges for σ > (½) and that (s−(½))Z_Q(s) can be continued to an entire function of s. Let σ be any constant with 0 < δ < 1/100. Then it is proved that Z_Q(s) has ≫_δTlogT zeros in the rectangle (|σ−½|≤δ, T≤t≤2T).