Bilinear forms in Weyl sums for modular square roots and applications

TL;DR

This paper improves bounds on the distribution of primes in quadratic fields and introduces a bilinear form estimate for Weyl sums related to modular square roots, with applications to prime distribution and equidistribution problems.

Contribution

It provides new lower bounds for primes splitting in quadratic fields and develops a novel bilinear form estimate for Weyl sums with power savings, connecting automorphic forms and number theory.

Findings

Improved bounds for the number of primes splitting in quadratic fields.

A new bilinear form estimate with power saving in the Pólya-Vinogradov range.

Applications to equidistribution of quadratic roots of primes and related conjectures.

Abstract

Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalently Sali\'{e} sums). Our estimate has a power saving in the so-called P{\'o}lya-Vinogradov range, and our methods involve studying an additive energy coming from quadratic residues in $\mathbb{F}_q$. This bilinear form is inspired by the recent automorphic motivation: the second moment for twisted $L$-functions attached to Kohnen newforms has recently been computed by the first and fourth authors. So the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form. These include the equidistribution of quadratic roots of primes, products of primes, and relaxations of a conjecture of Erdos-Odlyzko-Sarkozy.