Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes

TL;DR

The paper establishes nontrivial bounds for shifted bilinear sums involving Salié sums modulo a large prime $q$, and uses these to study the distribution of modular square roots of shifted primes $x^2\equiv ap+b\pmod{q}$ with $p\le P$. It proves sharp, range-dependent bounds for Type I/II sums, their smoothed and hyperbolic variants, and for sums over primes via Vaughan and Heath–Brown identities, combining Fourier-analytic methods, the Weil bound, and curve-exponential-sum estimates. A central theme is translating bilinear-sum bounds into distribution statements for shifted primes, yielding discrepancy bounds for the set of Salie-derived roots and asymptotic formulas in ranges tied to $q$. The results have potential applications to moments of Dirichlet $L$-functions and other questions in additive combinatorics and algebraic geometry over finite fields. Overall, the work extends the toolkit for bilinear sums with Salié and related sums, providing new bounds in regimes where $b\neq 0$ and shifted-prime problems are considered.

Abstract

We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Salié 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.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Corollary 2.8
• Remark 2.9
• Theorem 2.10