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 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).