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. 
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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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