Title: Exact Formulas for the Generalized Sum-of-Divisors Functions,
Abstract: We prove new exact formulas for the generalized sum-of-divisors functions.
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. 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 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 also consider applications of our new
results to asymptotic approximations for sums over these divisor functions and
to the forms of perfect numbers defined by the special case of the divisor
function, $\sigma(n)$, when $\alpha := 1$.
Keywords: divisor function; sum-of-divisors function; Lambert series; perfect
number.
MSC (2010): 30B50; 11N64; 11B83