# Bounds on prime gaps as a consequence of the divergence of the series of reciprocal primes

## TL;DR

This paper derives inequalities for prime gaps based on the divergence of the reciprocal primes series, linking prime gap bounds to classical divergence results and exploring implications for twin primes.

## Contribution

It introduces a new inequality for prime gaps derived from the divergence of reciprocal primes, connecting classical analysis with prime number theory.

## Key findings

- A general inequality for prime gaps holds infinitely often.
- The divergence of reciprocal primes implies bounds on prime gaps.
- Connections between prime gaps and twin prime conjecture are discussed.

## Abstract

In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.