On bilinear sums with modular square roots and applications III

We continue our investigations of bilinear sums with modular square roots and the large sieve for square moduli in our recent article "On bilinear sums with modular square roots and applications II", arXiv:2603.00768. In the present article, we focus on the case of prime square moduli for which our previous method in the said article did not yield any improvement. Now we modify this method to make progress for these moduli. The key idea is to restrict certain quadratic Gauss sums to reduced residue classes, which results in significant cancellations in certain cases.