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

Abstract

We obtain an asymptotic formula for the number of primes $p\leq x_1$, $p\leq x_2$ such that $p_1(p_2+a)\equiv l \pmod q$ with $(a,q)=(l,q)=1$, $q\leq x^{\kappa_0}$, $x_1\geq x^{1-\alpha}$, $x_2\geq x^{\alpha}$, $$ {\kappa}_0=\frac{1}{2.5+\theta+\varepsilon}, \quad \alpha\in \left[(\theta+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right], $$ where $\theta=1/2$, 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