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 $a,b\not\equiv 0\mathrm{mod}q$, the distribution ofsolutions to the congruence $x^2\equiv ap+b\mathrm{mod}q$, over primes $p\le P$. This is similar to the recently studied case of $b=0$, however the case $b\not\equiv 0\mathrm{mod}q$ exhibits some new difficulties.