On the distribution of modular square roots of primes

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences \(x^2 \equiv p \pmod q\) $x^2\equiv p(modq)$ with primes \(p\leqslant P\) $p⩽P$ and \(q \leqslant Q\) $q⩽Q$ . This can be considered as a combined scenario of Duke, Friedlander and Iwaniec with averaging only over the modulus q and of Dunn, Kerr, Shparlinski and Zaharescu with averaging only over p.