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A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis

Trey Smith

TL;DR

The paper addresses whether a generalized distribution principle for shifted correlations of the von Mangoldt function can imply the twin prime conjecture. It introduces GEH-2, a bilinear extension of the Elliott-Halberstam conjecture, bounding averages of the shifted-correlation error $E_2(x;q,a,h)$ over moduli up to $x^{\vartheta}$ with $0<\vartheta<2$. The key result shows that GEH-2 with $\vartheta>1$ and $h=2$ implies the asymptotic $\sum_{n\le x}\Lambda(n)\Lambda(n+2)=\mathfrak{S}(2)x+O(x/(\log x)^A)$ for any $A>0$, thereby proving the infinitude of twin primes. This work connects the twin-prime problem to bilinear distribution principles and motivates deeper study of shifted convolution structures and the singular-series framework within the Hardy-Littlewood k-tuple context.

Abstract

We propose a generalization of the Elliott-Halberstam conjecture concerning the distribution of prime pairs in arithmetic progressions. This conjecture, which we call the Generalized Elliott-Halberstam Conjecture for Shifted Convolutions (GEH-2), provides a level of distribution for correlations of the von Mangoldt function. We show that GEH-2 implies the twin prime conjecture, describe heuristic and analytic motivation, and discuss implications for prime gaps and $k$-tuple patterns.

A Generalized Elliott-Halberstam Conjecture Implying the Twin Prime Hypothesis

TL;DR

The paper addresses whether a generalized distribution principle for shifted correlations of the von Mangoldt function can imply the twin prime conjecture. It introduces GEH-2, a bilinear extension of the Elliott-Halberstam conjecture, bounding averages of the shifted-correlation error over moduli up to with . The key result shows that GEH-2 with and implies the asymptotic for any , thereby proving the infinitude of twin primes. This work connects the twin-prime problem to bilinear distribution principles and motivates deeper study of shifted convolution structures and the singular-series framework within the Hardy-Littlewood k-tuple context.

Abstract

We propose a generalization of the Elliott-Halberstam conjecture concerning the distribution of prime pairs in arithmetic progressions. This conjecture, which we call the Generalized Elliott-Halberstam Conjecture for Shifted Convolutions (GEH-2), provides a level of distribution for correlations of the von Mangoldt function. We show that GEH-2 implies the twin prime conjecture, describe heuristic and analytic motivation, and discuss implications for prime gaps and -tuple patterns.

Paper Structure

This paper contains 5 sections, 3 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Generalized Conjecture
  4. GEH-2 Implies the Twin Prime Conjecture
  5. Discussion and Further Directions

Key Result

Lemma 2.1

Let $h \in 2\mathbb{Z}$. Then where $\phi_2(q)$ is the number of residue classes $a \bmod q$ with $(a,q) = (a+h,q) = 1$. Moreover, under Möbius inversion with coprimality condition $(n(n+h),q)=1$, the sum converges to $1$.

Theorems & Definitions (10)

  • Conjecture 1.1: Elliott-Halberstam, EH
  • Lemma 2.1: Convergence of the Singular Series
  • proof : Sketch
  • Definition 3.1
  • Conjecture 3.2: Generalized Elliott-Halberstam for Shifted Convolutions (GEH-2)
  • Remark 3.3
  • Lemma 3.4: Error Control for Large Moduli
  • proof : Sketch
  • Theorem 4.1: Main implication
  • proof : Proof sketch