[Paper Review] Survey of Dirichlet Series of Multiplicative Arithmetic Functions

Richard J. Mathar|arXiv (Cornell University)|Jun 20, 2011

Analytic Number Theory Research24 references3 citations

TL;DR

This paper systematically derives Dirichlet series for multiplicative arithmetic functions by expressing them as products and ratios of Riemann zeta functions or infinite Euler products. It leverages Dirichlet convolution, multiplicative properties, and Bell series to classify and compute generating functions for key functions like Euler's totient, divisor functions, and higher-order Möbius and Jordan functions, providing closed-form expressions and OEIS A-numbers for reference.

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.

Motivation & Objective

• To systematically derive Dirichlet generating functions for important multiplicative arithmetic functions.
• To express these generating functions as finite or infinite products/ratios of Riemann zeta functions.
• To provide a unified framework using Bell series and Dirichlet convolution for deriving generating functions.
• To link results to OEIS sequences via A-numbers for cross-referencing and validation.
• To extend known results to higher-order multiplicative functions such as unitary Jordan and higher-order Möbius functions.

Proposed method

• Use of the Bell series representation: ∏ₚ ∑ₑ≥₀ a(pᵉ)/pᵉˢ to express Dirichlet series as Euler products.
• Application of Dirichlet convolution to inherit multiplicativity and derive generating functions for composite functions.
• Transformation of infinite Euler products into ratios of Riemann zeta functions using identities involving (1±pˡ⁻ᵘˢ)⁻¹.
• Use of master equations a(pᵉ) to define functions at prime powers and extend to all integers via multiplicativity.
• Derivation of generating functions for powers, divisor functions, totients, Möbius-type functions, and solutions to xᵗ≡0 mod n.
• Numerical evaluation via truncation of Euler product expansions to approximate ζ_D(s) in the complex plane.

Experimental results

Research questions

• RQ1 How can the Dirichlet series of multiplicative arithmetic functions be expressed in terms of Riemann zeta functions?
• RQ2 What is the role of Dirichlet convolution in generating new multiplicative functions and their associated Dirichlet series?
• RQ3 How do Bell series representations facilitate the derivation of infinite product forms for Dirichlet generating functions?
• RQ4 What are the closed-form expressions for Dirichlet series of higher-order Möbius and Jordan functions?
• RQ5 How do the convergence regions of these Dirichlet series relate to the exponents in the Euler product components?

Key findings

• The Dirichlet series for nᵏ is ζ(s−k), and for the constant function 1 it is ζ(s), with the Möbius function μ(n) generating 1/ζ(s).
• Euler’s totient function φ(n) has Dirichlet series ζ(s−1)/ζ(s), derived from the Dirichlet convolution φ = μ⋆id.
• The unitary Jordan function Jₖ⋆(n) has generating function ζ(s−k) × ∏ₚ (1−p⁻ˢ)(1+pᵏ⁻²ˢ)(1−p⁻²ˢ)(1+pᵏ⁻³ˢ)²⋯ for s>1+k.
• The number of solutions to xᵗ≡0 mod n is multiplicative with a(pᵉ)=p^{⌊(t−1)e/t⌋}, and its Dirichlet series is ∏ₚ (1 + ∑_{r=1}^{t−1} p^{r−1−rs}) / (1−p^{t−1−st}) for t≥2.
• The smallest x>0 such that xᵗ≡0 mod n has Dirichlet series ∏ₚ (1+p^{1−s})/(1−p^{1−2s}) = ζ(2s−1)ζ(s−1)/ζ(2s−2) for t=2.
• Higher-order Möbius functions μₖ(n) have Bell series (1−2p⁻ᵏˢ+p⁻⁽ᵏ⁺¹⁾ˢ)/(1−p⁻ˢ), and their Dirichlet series are expressed as infinite products involving zeta functions with specific exponent patterns.