Jacobsthal's function and a generalisation of Euler's totient

TL;DR

This paper introduces a generalized Euler's totient function to establish a tighter upper bound on Jacobsthal's function, improving understanding of the distribution of coprime integers within sequences.

Contribution

It proposes a novel generalization of Euler's totient function to derive a stronger bound on Jacobsthal's function h(k).

Findings

Derived a new upper bound on h(k) using the generalized totient.

Improved the theoretical understanding of coprime integer sequences.

Strengthened the connection between totient functions and Jacobsthal's function.

Abstract

Jacobsthal's function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to P_k, the product of the first k primes. The best known bound on h(k) is h(k) < C (k ln k)^2 for some unknown constant C, due to Iwaniec. We use a generalisation of Euler's totient function to give a stronger bound on h(k).