Distribution of products of shifted primes in arithmetic progressions with increasing difference
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910920479866880 |
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| author | Rakhmonov, Zarullo |
| author_facet | Rakhmonov, Zarullo |
| contents | 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 $(a,q)=(l,q)=1$, $q\leq x^{κ_0}$, $x_1\geq x^{1-α}$, $x_2\geq x^α$, $$ κ_0=\frac{1}{2.5+θ+\varepsilon}, \quad α\in \left[(θ+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right], $$ where $θ=1/2$, if $q$ is a cube free and $θ=\frac{5}{6}$ 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 |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of products of shifted primes in arithmetic progressions with increasing difference Rakhmonov, Zarullo Number Theory 11N05 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 $(a,q)=(l,q)=1$, $q\leq x^{κ_0}$, $x_1\geq x^{1-α}$, $x_2\geq x^α$, $$ κ_0=\frac{1}{2.5+θ+\varepsilon}, \quad α\in \left[(θ+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right], $$ where $θ=1/2$, if $q$ is a cube free and $θ=\frac{5}{6}$ 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 |
| title | Distribution of products of shifted primes in arithmetic progressions with increasing difference |
| topic | Number Theory 11N05 |
| url | https://arxiv.org/abs/2504.19316 |