Paper detail

Sums of generalized harmonic series for kids from five to fifteen

We examine the remarkable connection, first discovered by Beukers, Kolk and Calabi, between $ζ(2n)$, the value of the Riemann zeta-function at an even positive integer, and the volume of some $2n$-dimensional polytope. It can be shown that this volume is equal to the trace of a 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 $ζ(2n)$. In the case of odd positive integers, the expression for the kernel enables us to rediscover an integral representation for $ζ(2n+1)$, obtained by a different method by Cvijović and Klinowski. Finally, we indicate that the origin of the miraculous Beukers-Kolk-Calabi change of variables in the multidimensional integral, which is at the heart of this circle of ideas, can be traced to the amoeba associated with the certain Laurent polynomial. The paper is dedicated to the memory of Vladimir Arnold (1937-2010).