2006/07/20 by Daniel B. Grünberg, Grünberg, Daniel B.
Mathematics · #05A10 #11A07 #30B10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/0607514
openalex publication_date 2006/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums ∑k=1n (log k)p / kq, ~∑ kq (log k)p, ~∑ (log k)p /(n-k)q, ~∑ 1/kq (log k)p in closed form to arbitrary order (p,q ∈\N). The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either ζ(p)(± q) (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of ζ(p)(± q) faster than any current software. One of the constants also appears in the expansion of the function ∑n≥ 2 (nlog n)-s around the singularity at s=1; this requires the asymptotics of the incomplete gamma function. The manipulations involve polylogs for which we find a representation in terms of Nielsen integrals, as well as mysterious conjectures for Bernoulli numbers. Applications include the determination of the asymptotic growth of the Taylor coefficients of (-z/log(1-z))k. We also give the asymptotics of Stirling numbers of first kind and their formula in terms of harmonic numbers.