Asymptotic properties of integral points $(a_1,a_2)$, satisfying the congruence $a_1a_2\equiv l(q)$

V. A. Bykovskii

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 4, Tome 112 (1981), pp. 5-25

Résumé

The results of I. M. Vinogradov and van der Corput regarding the number of integral points under a curve are generalized to the case when on the integral points $(a_1,a_2)$ one imposes the additional condition $a_1a_2\equiv l(\operatorname{mod}q)$. A corollary is an asymptotic formula for $$ \sum^p_{z=1}\tau(z^2+D) $$ with the remainder $O(P^{5/6+\varepsilon})$ instead of Hooley's estimate $O(P^{8/9+\varepsilon})$. It is shown how with the aid of the spectral theory of automorphic functions one can bring the estimate to $O(P^{2/3+\varepsilon})$.