On Locally Repeated Values of Certain Arithmetic Functions, IV

Abstract

Let ω(n) denote the number of prime divisors of n and let Ω(n) denote the number of prime power divisors of n. We obtain upper bounds for the lengths of the longest intervals below x where ω(n), respectively Ω(n), remains constant. Similarly we consider the corresponding problems where the numbers ω(n), respectively Ω(n), are required to be all different on an interval. We show that the number of solutions g(n) to the equation m+ω(m)=n is an unbounded function of n, thus answering a question posed in an earlier paper in this series. A principal tool is a Turan-Kubilius type inequality for additive functions on arithmetic progressions with a large modulus.