math

(what is this?)

(what is this?)

# Title: On the sums Sum((4k+1)^(-n),k,-inf,+inf)

Abstract: 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.
 Comments: 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 Subjects: Classical Analysis and ODEs (math.CA); Combinatorics (math.CO) MSC classes: 11B68 (Primary) 11M06, 05A15 (Secondary) Journal reference: Amer. Math. Monthly_110_ #7 (Aug.-Sep. 2003), 561-573 Cite as: arXiv:math/0101168 [math.CA] (or arXiv:math/0101168v5 [math.CA] for this version)

## Submission history

From: Noam D. Elkies [view email]
[v1] Sat, 20 Jan 2001 17:57:04 GMT (12kb)
[v2] Thu, 25 Jan 2001 19:55:07 GMT (12kb)
[v3] Fri, 23 Aug 2002 02:53:41 GMT (13kb)
[v4] Mon, 4 Aug 2003 01:11:14 GMT (13kb)
[v5] Tue, 5 Aug 2003 15:32:31 GMT (13kb)
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8746108412742615, "perplexity": 2140.911079880782}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542932.99/warc/CC-MAIN-20161202170902-00412-ip-10-31-129-80.ec2.internal.warc.gz"}