On Character Sums of Binary Quadratic Forms

We establish character sum bounds of the form ∣∣∣∣ ∑ a≤x≤a+H b≤y≤b+H χ(x + ky) ∣∣∣∣ < p−τH2, where χ is a nontrivial character (mod p), p 1 4 +ε < H < p, and |a|, |b| < p H. As an application, we obtain that given k ∈ Z\{0}, x + k is a quadratic non-residue (mod p) for some 1 ≤ x < p 1 2e. Introduction. Let k be a nonzero integer. Let p be a large prime and let H ≤ p. We are interested in the character sum ∑ x,y χ(x 2 +ky), where χ (mod q) is a nontrivial character, and x and y run over intervals of length H; say a ≤ x ≤ a + H and b ≤ y ≤ b + H, and a and b are less than p H. The trivial bound for this character sum is H, and we seek an upper bound of the form H2p−δ for some δ > 0. Burgess [Bu3] considered such character sums, and obtained the desired H2p−δ estimate provided H ≥ p 13+2. Moreover, in the case that x + ky is irreducible (mod p) (i.e., −k is a quadratic non-residue (mod p)), Burgess obtained such cancelation in the wider range H ≥ p 14+2. In this paper we obtain a corresponding result in the case that x + ky is reducible (mod p) ( i.e., −k is a quadratic residue (mod p)). More precisely, we prove Theorem. Given ε > 0, there is τ > 0 such that if p is a sufficiently large prime and H is an integer satisfying p 1 4 +ε < H < p, (0.1) we have 2000 Mathematics Subject Classification.Primary 11L40, 11L26; Secondary 11A07, 11B75.