Papers › Exact Formulas for the Generalized Sum-of-Divisors Functions
Exact Formulas for the Generalized Sum-of-Divisors Functions
Maxie D. Schmidt
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We prove new exact formulas for the generalized sum-of-divisors functions, σ_α(x) := ∑_(d|x) d^α. The formulas for σ_α(x) when α∈ℂ is fixed and x ≥1 involves a finite sum over all of the prime factors n ≤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 ≤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, Φₙ(q), which completely factorize the Lambert series terms (1-qⁿ)⁻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 σ_α(n) and the summatory functions ∑_(n ≤x) σ_α(n). Keywords: divisor function; sum-of-divisors function; Lambert series; perfect number. MSC (2010): 30B50; 11N64; 11B83
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