 Online ISSN 1552-4485; Print ISSN 0033-569X

Evaluation of harmonic sums with integrals

Authors: Vivek Kaushik and Daniele Ritelli
Journal: Quart. Appl. Math. 76 (2018), 577-600
MSC (2010): Primary 52B11, 97K20, 97K50
DOI: https://doi.org/10.1090/qam/1499
Published electronically: February 6, 2018
MathSciNet review: 3805043
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Abstract: We consider the sums $S(k)=\sum _{n=0}^{\infty }\frac {(-1)^{nk}}{(2n+1)^k}$ and $\zeta (2k)=\sum _{n=1}^{\infty }\frac {1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show $S(2)=\pi ^2/8,$ which implies Euler’s identity $\zeta (2)=\pi ^2/6.$ Then, we generalize each integral in order to find the considered sums. The $k$ dimensional analogue of the first integral is the density function of the quotient of $k$ independent, nonnegative Cauchy random variables. In seeking this function, we encounter a special logarithmic integral that we can directly relate to $S(k).$ The $k$ dimensional analogue of the second integral, upon a change of variables, is the volume of a convex polytope, which can be expressed as a probability involving certain pairwise sums of $k$ independent uniform random variables. We use combinatorial arguments to find the volume, which in turn gives new closed formulas for $S(k)$ and $\zeta (2k).$ The $k$ dimensional analogue of the last integral, upon another change of variables, is an integral of the joint density function of $k$ Cauchy random variables over a hyperbolic polytope. This integral can be expressed as a probability involving certain pairwise products of these random variables, and it is equal to the probability from the second generalization. Thus, we specifically highlight the similarities in the combinatorial arguments between the second and third generalizations.

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Vivek Kaushik
Affiliation: Department of Mathematics, University of Illinois Urbana-Champaign, Champaign, Illinois 61820
Address at time of publication: 1409 West Green Street, Urbana, Illinois 61801
Email: vskaush2@illinois.edu

Daniele Ritelli
Affiliation: Department of Statistical Sciences, Università di Bologna, Bologna, Italy
Address at time of publication: Via delle Belle Arti 41 40126 Bologna Italy
MR Author ID: 618511
Email: daniele.ritelli@unibo.it

Keywords: Basel Problem, multiple integrals, random variables, polytope
Received by editor(s): October 10, 2017
Published electronically: February 6, 2018