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

Zarullo Rakhmonov

TL;DR

This work extends Karatsuba's method for ternary multiplicative problems to the distribution of products of shifted primes in arithmetic progressions with growing difference. It proves an asymptotic formula for the count of pairs $(p_1,p_2)$ with $p_1\le x_1$, $p_2\le x_2$ such that $p_1(p_2+a) \equiv l \pmod q$, under $q \le x^{\kappa_0}$ with $\kappa_0 = \frac{1}{2.5+\theta+\varepsilon}$ and $\theta=\frac{1}{2}$ if $q$ is cube-free (otherwise $\theta=\frac{5}{6}$). The main term is $\frac{1}{\varphi(q)} \prod_{p|q}(1-\frac{1}{p-1}) \mathrm{Li}(x_1)\mathrm{Li}(x_2)$ with a power-saving error, derived from average Chebyshev sums over Dirichlet characters and short sums of nonprincipal characters in shifted primes, together with Bombieri–Vinogradov and Brun–Titchmarsh-type bounds. The result generalizes Karatsuba's formula and tightens previous bounds by allowing a broader range for $q$ and more general moduli, highlighting the role of cube-free vs non-cube-free moduli via the parameter $\theta$.

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 $(a,q)=(l,q)=1$, $q\leq x^{κ_0}$, $x_1\geq x^{1-α}$, $x_2\geq x^α$, $$ κ_0=\frac{1}{2.5+θ+\varepsilon}, \quad α\in \left[(θ+\varepsilon)\frac{\ln q}{\ln x}, 1-2.5\frac{\ln q}{\ln x}\right], $$ 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

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

TL;DR

This work extends Karatsuba's method for ternary multiplicative problems to the distribution of products of shifted primes in arithmetic progressions with growing difference. It proves an asymptotic formula for the count of pairs with , such that , under with and if is cube-free (otherwise ). The main term is with a power-saving error, derived from average Chebyshev sums over Dirichlet characters and short sums of nonprincipal characters in shifted primes, together with Bombieri–Vinogradov and Brun–Titchmarsh-type bounds. The result generalizes Karatsuba's formula and tightens previous bounds by allowing a broader range for and more general moduli, highlighting the role of cube-free vs non-cube-free moduli via the parameter .

Abstract

We obtain an asymptotic formula for the number of primes , such that with , , , , where , if is a cube free and 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
Paper Structure (2 sections, 7 theorems, 43 equations)

This paper contains 2 sections, 7 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Teorema-RZKh-RPSdPrChVArProgr']}

Key Result

Theorem 1.1

Let $\varepsilon \in \left(0, \frac{1}{4}\right]$; $x \ge x_0(\varepsilon)$ be a sufficiently large positive number; let $q$ be a prime number such that $q \le x^{\ae_0}$, where $\kappa_0 = 1/(4.6 + \varepsilon)$; $(a, q) = 1$, $(l, q) = 1$; and let $\alpha$ be an arbitrary number in the interval where $x_1 \ge x^{1 - \alpha}$, $x_2 \ge x^{\alpha}$; $p_1$, $p_2$ are prime numbers; and let $\pi_2(

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Theorem 1.2
  • Lemma 1.5