Distribution of products of shifted primes in arithmetic progressions with increasing difference

We obtain an asymptotic formula for the number of primes $p\le x_1$, $p\le x_2$ such that $p_1(p_2+a)\equiv l(\mathrm{m}\mathrm{o}\mathrm{d}q)$ with $(a,q)=(l,q)=1$, $q\le x^{{\kappa }_0}$, $x_1\ge x^{1-\alpha }$, $x_2\ge x^{\alpha }$, $${\kappa }_0=\frac{1}{2.5+\theta +\epsilon },\alpha \in [(\theta +\epsilon )\frac{\ln q}{\ln x},1-2.5\frac{\ln q}{\ln x}],$$ where $\theta =1\mathrm{/}2$, if $q$ is a cube free and $\theta =\frac{5}{6}$ otherwise. This is the refinement and generalization of the well-known formula of A.~A.~Karatsuba.\ Keywords: {Dirichlet character, shifted primes, short sum of characters with primes}\ Bibliography: 39 references