Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
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.
Fields of Study & Subjects
Analytic Number Theory Research
Limits and Structures in Graph Theory
Algebraic Geometry and Number Theory
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 15 January 2026
- Document type
- Preprint