Geodesic
Distribution of products of shifted primes in arithmetic progressions with increasing difference
Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 156-168

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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 $q\leq x^{\mathrm{ae}_0}$, $x_1\geq x^{1-\alpha}$, $x_2\geq x^{\alpha}$, $$\mathrm{ae}_0=\frac{1}{2.5+\theta+\varepsilon}, \alpha\in \left[(\theta+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right],$$ 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. 
Z. Kh. Rakhmonov. Distribution of products of shifted primes in arithmetic progressions with increasing difference. Čebyševskij sbornik, Tome 23 (2022) no. 3, pp. 156-168. http://geodesic.mathdoc.fr/item/CHEB_2022_23_3_a10/
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