Negative moments of the gaps between consecutive primes

TL;DR

This paper develops heuristic formulas for negative moments of prime gaps, including an analytical approximation for the sum of reciprocals of these gaps, supported by extensive computational data up to 4×10^{18}.

Contribution

It introduces new heuristic formulas for negative moments of prime gaps and provides an analytical approximation for the sum of reciprocals of these gaps.

Findings

Heuristic formulas for negative moments of prime gaps.

Analytical approximation for the sum of reciprocals of prime gaps.

Extensive computational validation up to 4×10^{18}.

Abstract

We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $\pi(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{\pi^2(x)}{x-2\pi(x)}\log\Big(\frac{x}{2\pi(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.