# On the moments of the gaps between consecutive primes

## TL;DR

This paper heuristically derives formulas for the moments of gaps between consecutive primes less than x, relating them to prime counting functions, and supports findings with computational data.

## Contribution

It introduces a heuristic formula for the moments of prime gaps expressed via prime counting functions, providing a new perspective on prime gap distribution.

## Key findings

- Derived a formula for the k-th moments of prime gaps.
- Validated the formula with computational data.
- Connected moments of gaps to prime counting functions.

## Abstract

We derive heuristically formula for the $k$--moments $M_k(x)$ of the gaps between consecutive primes$<x $ represented directly by $x$$\pi(x)$ --- the number of primes up to: $M_k(x)= \Gamma(k+1)x^k/\pi^{k-1}(x)+\mathcal{O}(x)$, We illustrate obtained results by computer data.