Sums of generalized harmonic series for kids from five to fifteen

Details Sums of generalized harmonic series for kids from five to fifteen Sums of generalized harmonic series for kids from five to fifteen

2010-03-16

arxiv.org/abs/1003.3602v1

Mathematics

Classical Analysis and ODEs

18 pages, no figures

Scientific paper

We reexamine remarkable connection, first discovered by Beukers, Kolk and Calabi, between $\zeta(2n)$, the value of the Riemann zeta-function at even positive integer, and the volume of some $2n$-dimensional polytope. It can be shown that this volume equals to the trace of some compact self-adjoint operator. We provide an explicit expression for the kernel of this operator in terms of Euler polynomials. This explicit expression makes it easy to calculate the volume of the polytope and hence $\zeta(2n)$. In the case of odd positive integers, the expression for the kernel enables to rediscover an integral representation for $\zeta(2n+1)$, obtained originally by different method by Cvijovic and Klinowski. Finally, we indicate that the origin of the Beukers-Kolk-Calabi's miraculous change of variables in the multidimensional integral, which is at the heart of all of this business, can be traced down to the amoeba associated with the certain Laurent polynomial.

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