Exact Formulas for the Generalized Sum-of-Divisors Functions

Abstract

We prove new exact formulas for the generalized sum-of-divisors functions, ${\sigma }_{\alpha }(x):={\sum }_{d|x}{d}^{\alpha }$. The formulas for ${\sigma }_{\alpha }(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x\ge 1$ involves a finite sum over all of the prime factors $n\le x$ and terms involving the $r$-order harmonic number sequences and the Ramanujan sums $c_d(x)$. The generalized harmonic number sequences correspond to the partial sums of the Riemann zeta function when $r>1$ and are related to the generalized Bernoulli numbers when $r\le 0$ is integer-valued. A key part of our new expansions of the Lambert series generating functions for the generalized divisor functions is formed by taking logarithmic derivatives of the cyclotomic polynomials, ${\Phi }_n(q)$, which completely factorize the Lambert series terms $(1-q^n{)}^{-1}$ into irreducible polynomials in $q$. We focus on the computational aspects of these exact expressions, including their interplay with experimental mathematics, and comparisons of the new formulas for ${\sigma }_{\alpha }(n)$ and the summatory functions ${\sum }_{n\le x}{\sigma }_{\alpha }(n)$. Keywords: divisor function; sum-of-divisors function; Lambert series; perfect number. MSC (2010): 30B50; 11N64; 11B83

Added a new theorem for asymptotics of the average orders of certain sum-of-divisors functions with small error term

References in corpus (3)

• Mertens' Proof of Mertens' Theorem
• A Partition Identity Related to Stanley's Theorem
• An Explicit Formula For The Divisor Function

Cited by in corpus (1)

• A catalog of interesting and useful Lambert series identities