Negative moments of the gaps between consecutive primes

Marek Wolf

Published 2018-05-13Version 1

We derive heuristically 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 a closed 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 $x=4\times 10^{18}$.