Distribution of products of shifted primes in arithmetic progressions with increasing difference
https://doi.org/10.22405/2226-8383-2022-23-3-156-168
Abstract
We obtain an asymptotic formula for the number of primes 𝑝 ≤ 𝑥1, 𝑝 ≤ 𝑥2 such that
𝑝1(𝑝2 + 𝑎) ≡ 𝑙 (mod 𝑞) with 𝑞 ≤ 𝑥æ0 , 𝑥1 ≥ 𝑥1−𝛼, 𝑥2 ≥ 𝑥𝛼,
$$æ0 =1/(2, 5 + 𝜃 + 𝜀), 𝛼 ∈[︂(𝜃 + 𝜀)ln 𝑞/ln 𝑥, 1 − 2, 5(ln 𝑞/ln 𝑥)]︂,$$
where 𝜃 = 1/2, if 𝑞 is a cube free and 𝜃 = 5
6 otherwise. This is the refinement and generalization
of the well-known formula of A.A.Karatsuba.
About the Author
Zarullo Khusenovich Rakhmonov
A. Dzhuraev Institute of Mathematics
Tajikistan
Tajikistan
doctor of physical and mathematical sciences, professor,
academician of the National Academy of Sciences of Tajikistan, director of the A. Dzhuraev Institute of Mathematics
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Review
For citations:
Rakhmonov Z.Kh. Distribution of products of shifted primes in arithmetic progressions with increasing difference. Chebyshevskii Sbornik. 2022;23(3):156-168. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-3-156-168
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