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 ≡ p q with primes p⩽ P and 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.