Exact Formulas for the Generalized Sum-of-Divisors Functions

TL;DR

This paper derives new exact formulas for generalized sum-of-divisors functions involving prime factors, harmonic numbers, and Ramanujan sums, with a focus on computational applications and mathematical properties.

Contribution

It introduces novel formulas for alpha(x) using prime factor sums, harmonic numbers, and cyclotomic polynomial derivatives, advancing the understanding of divisor functions.

Findings

New explicit formulas for alpha(x) involving prime factors and harmonic numbers

Connections between harmonic number sequences and Riemann zeta or Bernoulli numbers

Enhanced computational methods for divisor and summatory 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 $x \geq 1$ involves a finite sum over all of the prime factors $n \leq x$ 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 $r \leq 0$ 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 $\sum_{n \leq x} \sigma_{\alpha}(n)$. Keywords: divisor function; sum-of-divisors function; Lambert series; perfect number. MSC (2010): 30B50; 11N64; 11B83