Improved Upper Bound on Brun's Constant Under GRH

TL;DR

This work establishes a rigorous upper bound $B<2.1594$ for Brun's constant under the Generalized Riemann Hypothesis by combining a Selberg sieve-based bound for the twin-prime count $\pi_2(x)$ with a tail estimate for $B$ via intermediate sums $B(m_i,m_{i+1})$ and partial summation. The method relies on explicit sieve constants, a three-regime bound for $V(z)$, and GRH-based controls of error terms $E(x;d)$, all implemented with interval arithmetic for rigor. A preliminary bound $B(2,4\cdot 10^{18})\le 1.840518$ is supplemented by a parameter-optimized tail across many intervals, yielding the final GRH-based bound and a reproducible computational framework. The results demonstrate a rigorous approach to tightening constants in convergent sums over twin primes and provide a platform for further refinements and larger-x calculations.

Abstract

Brun's constant is the summation of the reciprocals of all twin primes, given by $B=\sum_{p \in P_2}{\left( \frac{1}{p} + \frac{1}{p+2}\right)}$. While rigorous unconditional bounds on $B$ are known, we present the first rigorous bound on Brun's constant under the GRH assumption, yielding $B < 2.1594$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.1
• Proposition 2.1
• proof
• Lemma 2.2
• Theorem 2.2
• proof
• Lemma 3.1
• proof