On the asymptotic density of the support of a Dirichlet convolution

TL;DR

This paper proves that the support of a Dirichlet convolution involving a multiplicative function with positive density support also has positive asymptotic density, extending previous results to more general functions.

Contribution

It generalizes prior work by establishing positive density of the convolution support for a broader class of functions, including non-multiplicative ones.

Findings

Support of Dirichlet convolution has positive density under specified conditions

Quantitative bounds are provided when f is multiplicative

Extends previous results on M"obius and Dirichlet transforms

Abstract

Let v be a multiplicative arithmetic function with support of positive asymptotic density. We prove that for any not identically zero arithmetic function f such that \sum_{f(n) \neq 0} 1 / n < \infty, the support of the Dirichlet convolution f * v possesses a positive asymptotic density. When f is a multiplicative function, we give also a quantitative version of this claim. This generalizes a previous result of P. Pollack and the author, concerning the support of M\"obius and Dirichlet transforms of arithmetic functions.