Maximum distance between consecutive primes and other related questions

TL;DR

This paper investigates the asymptotic behavior of prime distributions related to primorials, explores bounds on prime gaps, and provides an algorithm for computing the Jacobstal function to support conjectures on prime gaps.

Contribution

It introduces a new approach to analyze prime gaps using the Jacobstal function and offers an algorithm for its computation, advancing understanding of prime distribution asymptotics.

Findings

Derived asymptotic relations involving primes and primorials

Established bounds and conjectures on maximum prime gaps

Provided an algorithm for calculating the Jacobstal function

Abstract

The paper considers the asymptotic of the ratio of the number of primes not exceeding the primorial and the number of residues in the reduced system of residues for the given primorial. We study the relationship between asymptotic lower bounds for the values of the Jacobstal function and the maximum distance between successive primes. One algorithm for computing the Jacobstal function is given. The substantiation of the conjectures about the upper estimate of the maximum distance between successive prime numbers is given.