Exact Formulas for the Generalized Sum-of-Divisors Functions

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

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