On the distribution in the arithmetic progressions of reducible quadratic polynomials in short intervals

Abstract

In this paper we study the distribution in the arithmetic progressions (modulo a product of two primes) of reducible quadratic polynomials $(an+b)(cn+d)$ in short intervals, i.e. when $n \in [x, x + H], H = o(x)$; here $H = x^{\vartheta}$, with $\vartheta \in ]3/4, 1[$. Using Large Sieve techniques we get results beyond the classical level $\vartheta$, reaching $3\vartheta - 3/2$; these also improve the results of Salerno and Vitolo [6] in “large” intervals $(\vartheta = 1)$ obtaining level $3/2$ instead of $4/3$.