On Goldbach numbers in short intervals

TL;DR

Under the Riemann Hypothesis, the paper proves that for every $x\ge 2$ there is a Goldbach number in the short interval $(x, x+123\log^2 x]$, improving the previous constant. The authors bound the Selberg-type mean square $J_\theta(x,\delta)$ by connecting it to $J_\psi(x,\delta)$ through Saffari–Vaughan averaging and explicit zeta-zero analysis, and then control the error terms with Fourier-analytic and explicit-mean-value techniques. The key contributions are explicit constants $2.2258$ for $J_\psi$ and $2.5571$ for $J_\theta$, which enable a sharp conditional result on Goldbach numbers and demonstrate the power of explicit RH-based zero-analysis in short-interval problems. The work advances understanding of the distribution of prime sums and refines the conditional bounds for Goldbach representations with potential for further improvements in the constant via refined zero-analysis.

Abstract

Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent work of Cully-Hugill and Dudek, who obtained the constant $9696$ instead of $123$.

Theorems & Definitions (20)

• Theorem 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5