[Paper Review] Some results on Theory of Infinite Series and Divisor Sums

Nikos Bagis|arXiv (Cornell University)|Dec 24, 2009

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper presents novel formulas for infinite series and divisor sums using arithmetic functions, with applications to Jacobi elliptic theta functions. It derives product and sum identities for general arithmetic functions and applies them to modular forms, yielding new representations of theta functions through analytic number theory techniques.

ABSTRACT

In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the study of Jacobi elliptic theta functions theory.

Motivation & Objective

• To develop general product and sum formulas for arithmetic functions.
• To explore connections between infinite series and divisor sums in number theory.
• To apply derived formulas to the theory of Jacobi elliptic theta functions.
• To establish new analytic representations of modular forms via arithmetic identities.
• To extend techniques in generalized Dirichlet series and multiplicative number theory.

Proposed method

• Derives general product and sum identities for arithmetic functions using generating functions and Dirichlet series.
• Applies transformation techniques to relate divisor sums to infinite series expansions.
• Uses properties of multiplicative arithmetic functions to construct closed-form expressions.
• Leverages known identities in modular forms to connect results to Jacobi theta functions.
• Employs analytic number theory tools, including generating functions and series convergence analysis.
• Validates results through symbolic manipulation and consistency checks with known number-theoretic identities.

Experimental results

Research questions

• RQ1 How can general arithmetic functions be used to construct new infinite series and product formulas?
• RQ2 What is the relationship between divisor sums and infinite series in the context of multiplicative functions?
• RQ3 Can these identities be applied to derive new representations of Jacobi theta functions?
• RQ4 How do these formulas relate to known modular forms and their transformation properties?
• RQ5 What are the convergence and analytic properties of the derived series and products?

Key findings

• The paper establishes a general framework for infinite series and product formulas involving arithmetic functions.
• It derives new identities connecting divisor sums to generating functions of multiplicative arithmetic functions.
• The results yield explicit representations of Jacobi theta functions through infinite products and series.
• The derived formulas are consistent with known modular transformation laws of theta functions.
• The method enables systematic generation of identities for a broad class of arithmetic functions.
• The work provides a unified approach to studying divisor sums and infinite series via analytic number theory.