Abstract
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 (a1,a2) one imposes the additional condition a1,a2 ≡ l(mod q). A corollary is an asymptotic formula for $$\sum\limits_{z = 1}^p {\tau \left( {z^2 + D} \right)} $$ with the remainder O(P5/6+e) instead of Hooley's estimate O(P8/9+e). It is shown how with the aid of the spectral theory of automorphic functions one can bring the estimate to O(P2/3+e).
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