Accurate double inequalities for generalized harmonic numbers

Vito Lampret

Applied Mathematics and Computation, 2015, vol. 265, issue C, 557-567

Abstract: For n∈N and p∈R the nth harmonic number of order pH(n,p):=∑k=1n1kpis expressed in the form H(n,p)=H˜q(m,n,p)+Rq(m,n,p)where m,q∈N are parameters controlling the magnitude of the error term. The function H˜q(m,n,p) consists of m+2q+1 simple summands and the remainder Rq(m, n, p) is estimated, for p ≥ 0, as 0≤(−1)q+1Rq(m,n,p)<1π(1−2·4−q)(p2+q−1)πm)2q−1·1mp.Similar result is obtained also for p < 0 and for real zeta function (n=∞,p > 1) as well.

Keywords: Approximation; Estimate; Euler–Maclaurin summation; Generalized harmonic number; p-series; Zeta-generalized-Euler-constant function (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:265:y:2015:i:c:p:557-567

DOI: 10.1016/j.amc.2015.04.128

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