Clustering of linear combinations of multiplicative functions

A real-valued arithmetic function F is said to cluster about the point u is an element of R if the upper density of n with u - delta < F(n) < u is bounded away from 0, uniformly for all delta > 0. We establish a simple-to-check sufficient condition for a linear combination of multiplicative functions to be nonclustering, meaning not clustering anywhere. This provides a means of generating new families of arithmetic functions possessing continuous distribution functions. As a specific application, we resolve a problem posed recently by Luca and Pomerance. (C) 2017 Elsevier Inc. All rights reserved.