Arithmetic functions and their linear dependence

Ponpetch, Kanet; Laohakosol, Vichian; Mavecha, Sukrawan

doi:10.1063/1.5012173

Arithmetic functions and their linear dependence

Author(s)

Ponpetch, Kanet

Laohakosol, Vichian

Mavecha, Sukrawan

Date Issued

November 22, 2017

Type

Conference Paper

DOI

10.1063/1.5012173

Abstract

An arithmetic function is a complex-valued function defined over the positive integers. The set of arithmetic functions equipped with the usual addition and Dirichlet convolution forms a unique factorization domain. Complementing the results in the works of Komatsu et al., the problem of linear dependence over the complex field of arithmetic functions is investigated. Two general criteria are proved. Emphases are placed upon arithmetic functions which are solutions of additive equation, multiplicative equation, exponential equation and logarithmic equation. It is found that i) additive functions are always linearly dependent, ii) exponential functions are always linearly independent, and iii) the situation for logarithmic and multiplicative functions are more complex and conditions for their (in)dependence are derived.

Citation

Aip Conference Proceedings, 1905, 2017