Shifted bilinear sums of Sali\'e sums and the distribution of modular square roots of shifted primes

TL;DR

This paper derives bounds on shifted bilinear sums involving Salié sums modulo large primes and applies these to analyze the distribution of modular square roots of shifted primes, revealing new challenges when the shift is non-zero.

Contribution

It introduces new bounds on shifted bilinear sums with Salié sums and applies them to study the distribution of solutions to quadratic congruences over primes, especially for non-zero shifts.

Findings

Established bounds on Type-I and Type-II shifted bilinear sums with Salié sums.

Analyzed the distribution of solutions to x^2 ≡ ap + b mod q over primes p.

Identified new difficulties in the case when b ≠ 0.

Abstract

We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Sali\'e sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\not \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\not \equiv 0 \bmod q$ exhibits some new difficulties.