Jacobsthal's function and a generalisation of Euler's totient
Abstract
Jacobsthal's function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to Pk, 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).
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