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

Full-text PDF

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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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References
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*De summis serierum reciprocarum*, Commentarii academiae scientiarum Petropolitanae **7** (1740), no. 1740, 123–134.
- Tom M. Apostol,
*A proof that Euler missed: evaluating $\zeta (2)$ the easy way*, Math. Intelligencer **5** (1983), no. 3, 59–60. MR **737691**
- Dan Kalman,
*Six ways to sum a series*, College Math. J. **24** (1993), no. 5, 402–421. MR **3287877**
- Daniele Ritelli,
*Another proof of $\zeta (2)=\frac {\pi ^2}{6}$ using double integrals*, Amer. Math. Monthly **120** (2013), no. 7, 642–645. MR **3096470**
- Luigi Pace,
*Probabilistically proving that $\zeta (2)=\pi ^2/6$*, Amer. Math. Monthly **118** (2011), no. 7, 641–643. MR **2826455**
- Paul Bourgade, Takahiko Fujita, and Marc Yor,
*Euler’s formulae for $\zeta (2n)$ and products of Cauchy variables*, Electronic Communications in Probability **12** (2007), 73–80., DOI https://doi.org/10.1214/ECP.v12-1244
- Junesang Choi,
*Evaluation of certain alternating series*, Honam Math. J. **36** (2014), no. 2, 263–273. MR **3235500**
- James D. Harper,
*Another simple proof of $1+\frac {1}{2^2}+\frac {1}{3^2}+\dots =\frac {\pi ^2}{6}$*, Amer. Math. Monthly **110** (2003), no. 6, 540–541. MR **1984408**
- Nick Lord,
*Yet Another Proof That $\sum \frac {1}{n^{2}}=\frac {1}{6}\pi ^{2}$*, The Mathematical Gazette **86** (2002), no. 507, 477-479.
- Marek Capiński and Peter E. Kopp,
*Measure, Integral and Probability 2nd edition*, Springer, 2004.
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*The algebra of random variables*, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Wiley Series in Probability and Mathematical Statistics. MR **519342**
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*Table errata: **Table of integrals, series, and products* [corrected and enlarged edition, Academic Press, New York, 1980; MR 81g:33001] by I. S. Gradshteyn [I. S. Gradshteĭn] and I. M. Ryzhik, Math. Comp. **39** (1982), no. 160, 747–757. MR **669666**
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*Sums of generalized harmonic series and volumes*, Nieuw Arch. Wisk. (4) **11** (1993), no. 3, 217–224. MR **1251484**
- F. M. S. Lima,
*New definite integrals and a two-term dilogarithm identity*, Indag. Math. (N.S.) **23** (2012), no. 1-2, 1–9. MR **2877396**
- Joseph D’Avanzo and Nikolai A. Krylov,
*$\zeta (n)$ via hyperbolic functions*, Involve **3** (2010), no. 3, 289–296. MR **2739520**
- Noam D. Elkies,
*On the sums $\sum ^\infty _{k=-\infty }(4k+1)^{-n}$*, Amer. Math. Monthly **110** (2003), no. 7, 561–573. MR **2001148**
- Zurab Silagadze,
*Sums of generalized harmonic series for kids from five to fifteen*, arXiv preprint arXiv:1003.3602 (2010).
- Zurab Silagadze,
*Comment on the sums $S(n)=\sum ^\infty _{k=-\infty }(4k+1)^{-n}$*, Georgian Math. J. **19** (2012), no. 3, 587–595. MR **2984507**
- Richard P. Stanley,
*Two poset polytopes*, Discrete Comput. Geom. **1** (1986), no. 1, 9–23. MR **824105**
- Maxim Kontsevich and Don Zagier,
*Periods*, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 771–808. MR **1852188**

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Additional Information

**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

Article copyright:
© Copyright 2018
Brown University