Let $\nu$ 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 $f * \nu$ 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.

On the asymptotic density of the support of a Dirichlet convolution

SANNA, CARLO
2014-01-01

Abstract

Let $\nu$ 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 $f * \nu$ 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.
2014
134
1
12
https://arxiv.org/abs/1303.5617
Asymptotic density; Dirichlet convolution; Möbius inversion; Möbius transform; Natural density; Primary; Secondary; Sets of multiples; Uncertainty principle; Algebra and Number Theory
Sanna, Carlo
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1622112
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