Rapidly Convergent Series of the Divisors Functions

TL;DR

This paper presents new rapidly convergent series for the sum of divisors functions, enabling precise evaluations, estimates, and existence proofs of special values, with broad applications in number theory.

Contribution

It introduces novel rapidly convergent series representations for the sums of divisors functions, enhancing computational efficiency and theoretical analysis.

Findings

New series enable exact evaluations of power series

Series facilitate improved estimates of divisor sums

Applications include proving existence of special divisor function values

Abstract

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.