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arXiv · 2601.10113

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

Igor E. Shparlinski·Yixiu Xiao
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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.

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Igor E. Shparlinski, Yixiu Xiao. 2026-01-15. Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes. https://arxiv.org/abs/2601.10113

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