Survey of Dirichlet Series of Multiplicative Arithmetic Functions

2011/06/20 by Richard J. Mathar, Mathar, Richard J. · 1 citation
Mathematics · #11K65 #11M41 (Secondary) #11Y70 (Primary) 30B50 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT #msc:11K65 #msc:11M41 #msc:11Y70 #msc:30B50

paper · pdf · doi:10.48550/arxiv.1106.4038

Reorganized chapters 3.13 and 3.14. Updated bibliography

openalex publication_date 2011/06/20 · arxiv created 2012/07/04 · arxiv updated 2012/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading factors of the infinite product over zeta-functions. If rooted at the Dirichlet series for powers, for sums-of-divisors and for Euler's totient, the inheritance of multiplicativity through Dirichlet convolution or ordinary multiplication of pairs of arithmetic functions generates most of the results.

Cited by

• Sums involving the number of distinct prime factors function

Related

• Dirichlet series for complex powers of the Riemann zeta function
• Dirichlet product of derivative arithmetic with an arithmetic function multiplicative
• Refactorisation of the Dirichlet convolution
• Formalizing zeta and L-functions in Lean
• Factorisation of the product of Dirichlet series of completely\n multiplicative functions