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Primes in tuples and Romanoff's theorem

Artyom Radomskii

TL;DR

This work proves a quantitative lower bound for the number of primes in tuples of linear forms and derives Romanoff-type representation consequences. It builds on the Maynard sieve framework, augmented by well-distributed set hypotheses and Bombieri–Vinogradov-type distribution controls, to show that for $x\ge10$ and modest growth of $k$, there exists a prime $p_0$ such that, for a family of admissible linear forms $L_i(n)=a_i n+b_i$ with $(a_i,b_i)=1$, $a_i\le \exp(c_0\sqrt{\ln x})$, $(a_i,p_0)=1$ and $b_i\le x\exp(c_0\sqrt{\ln x})$, one has a positive lower bound on the count of $n\in[x,2x)$ for which at least $C^{-1}\ln k$ of the $L_i(n)$ are prime, namely at least $x/(\ln x)^k\exp(Ck)$. The authors extend the result to a related setting (Theorem T2) via a linear transformation and obtain Romanoff-type density estimates for representation functions $f_{\\mathcal{A}}(n)$. Central to the argument are three preparatory lemmas that produce an admissible subfamily, establish a BV-type distribution bound around a carefully chosen prime $B$, and verify the required hypotheses for the sieve framework. Together, these results advance quantitative understanding of primes in linear patterns and have implications for Romanoff-type representations, connecting sieve methods with density questions for primes in arithmetic structures.

Abstract

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.

Primes in tuples and Romanoff's theorem

TL;DR

This work proves a quantitative lower bound for the number of primes in tuples of linear forms and derives Romanoff-type representation consequences. It builds on the Maynard sieve framework, augmented by well-distributed set hypotheses and Bombieri–Vinogradov-type distribution controls, to show that for and modest growth of , there exists a prime such that, for a family of admissible linear forms with , , and , one has a positive lower bound on the count of for which at least of the are prime, namely at least . The authors extend the result to a related setting (Theorem T2) via a linear transformation and obtain Romanoff-type density estimates for representation functions . Central to the argument are three preparatory lemmas that produce an admissible subfamily, establish a BV-type distribution bound around a carefully chosen prime , and verify the required hypotheses for the sieve framework. Together, these results advance quantitative understanding of primes in linear patterns and have implications for Romanoff-type representations, connecting sieve methods with density questions for primes in arithmetic structures.

Abstract

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.
Paper Structure (6 sections, 9 theorems, 106 equations)

This paper contains 6 sections, 9 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Well-distributed sets
  4. Preparatory Lemmas
  5. Proof of Theorem \ref{['T1']} and Theorem \ref{['T2']}
  6. Acknowledgements

Key Result

Theorem 1.1

Let $0<\alpha< 1/4$ and $\beta > 0$. Then there exist a positive constant $C_{0}$ depending only on $\alpha$ such that for sufficiently large number $x$ and $C_{0} \leq k \leq (\ln x)^{\alpha}$, if $\{a_{1}n+b_{1},\ldots, a_{k}n+b_{k}\}$ is a set of $k$ distinct linear functions with $(a_{i}, b_{i}) Here $P^{+}(n)$ denotes the largest prime factor of $n$.

Theorems & Definitions (17)

  • Theorem 1.1: Chen.Ding
  • Theorem 1.2
  • proof : Deduction of Theorem \ref{['T.Ch.Ding']} from Theorem \ref{['T1']}
  • Theorem 1.3
  • Corollary 1.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 7 more