On the sums Sum((4k+1)^(-n),k,-inf,+inf)
| dc.creator | Elkies, Noam D. | |
| dc.date | 2001-01-20 | |
| dc.date | 2003-08-05 | |
| dc.date.accessioned | 2026-07-07T04:39:44Z | |
| dc.date.available | 2026-07-07T04:39:44Z | |
| dc.description | The sum in the title is a rational multiple of pi^n for all integers n=2,3,4,... for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi: the sum is written as a definite integral over the unit n-cube, then transformed into the volume of a polytope Pi_n in R^n whose vertices' coordinates are rational multiples of pi. We review Calabi's proof, and give two further interpretations. First we define a simple linear operator T on L^2(0,pi/2), and show that T is self-adjoint and compact, and that Vol(Pi_n) is the trace of T^n. We find that the spectrum of T is {1/(4k+1) : k in Z}, with each eigenvalue 1/(4k+1) occurring with multiplicity 1; thus Vol(Pi_n) is the sum of the n-th powers of these eigenvalues. We also interpret Vol(Pi_n) combinatorially in terms of the number of alternating permutations of n+1 letters, and if n is even also in terms of the number of cyclically alternating permutations of n letters. We thus relate these numbers with S(n) without the intervention of Bernoulli and Euler numbers or their generating functions. | |
| dc.description | 14 pages. A partly expository article, now published in the American Math. Monthly. Third revision, to include publication data and correct a few trivial typos. Includes the Abstract, which did not appear in the Monthly | |
| dc.identifier | https://arxiv.org/abs/math/0101168 | |
| dc.identifier | http://arxiv.org/abs/math/0101168 | |
| dc.identifier | Amer. Math. Monthly_110_ #7 (Aug.-Sep. 2003), 561-573 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60787 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Combinatorics | |
| dc.subject | 11B68 (Primary) 11M06, 05A15 (Secondary) | |
| dc.title | On the sums Sum((4k+1)^(-n),k,-inf,+inf) | |
| dc.type | text |