L-functions and character sums for quadratic forms (I)

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 c...

Large Character Sums: Burgess’s Theorem and Zeros of L-functions

We study the conjecture that ∑ n≤x χ(n) = o(x) for any primitive Dirichlet character χ (mod q) with x ≥ q , which is known to be true if the Riemann Hypothesis holds for L(s, χ). We show that it holds under the weaker assumption that ‘100%’ of the zeros of L(s, χ) up to height 14 lie on the critical line. We also establish various other consequences of having large character sums; for example, ...

Quadratic class numbers and character sums

We present an algorithm for computing the class number of the quadratic number field of discriminant d. The algorithm terminates unconditionally with the correct answer and, under the GRH, executes in Oε(|d|) steps. The technique used combines algebraic methods with Burgess’ theorem on character sums to estimate L(1, χd). We give an explicit version of Burgess’ theorem valid for prime discrimin...

Quadratic Functions, Optimization, and Quadratic Forms

Gauss Sums & Representation by Ternary Quadratic Forms

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...