{"id":57,"date":"2008-05-30T19:47:18","date_gmt":"2008-05-30T18:47:18","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/2008\/05\/30\/an-exercise\/"},"modified":"2024-02-20T13:40:59","modified_gmt":"2024-02-20T11:40:59","slug":"an-exercise","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2008\/05\/30\/an-exercise\/","title":{"rendered":"An exercise"},"content":{"rendered":"

Here is a striking example of the unfortunate (but probably unavoidable) dispersion of mathematics: at the end of the abstract of V. Arnold’s talk<\/a> at the recent conference in honor of A. Douady, one can read:<\/p>\n

The Cesaro mean values K\u2560\u00e9 of the numbers K(n) tend, as n tends to \u00d4\u00ea\u00d7, to a finite limit K\u2560\u00e9(\u00d4\u00ea\u00d7)=lim 1\/n \u00d4\u00ea\u00e6m=1<\/sub>n<\/sup> K(m) = 15\/\u00a4\u00c72<\/sup>. This theorem, deduced from the empirical observation of the coincidence of 20 first digits, is now proved, using the formula K\u2560\u00e9(\u00d4\u00ea\u00d7)=\u256c\u00c2(2)\/\u256c\u00c2(4)<\/p><\/blockquote>\n

Here, K(n) is defined before in the abstract as the expression<\/p>\n

$latex K(n)=\\prod_{p\\mid n}{(1+1\/p)}$<\/p>\n

Arnold’s achievements as mathematician are about as impressive as it can get. But the statement here is a completely elementary exercise in analytic number theory, and has been for at least one century (i.e., Dirichlet, or Chebychev, could do it in a few minutes, if not Euler). Here’s the proof in Chebychev style:<\/p>\n

$latex K(n)=\\sum_{d\\mid n}{\\mu(d)^2\/d}$<\/p>\n

hence, exchaning the sum over n<\/em> and the sum over d<\/em>, we get<\/p>\n

$latex \\sum_{n<X}{K(n)}=\\sum_{d<X}{\\mu(d)^2d^{-1} [X\/d]}$<\/p>\n

and replacing the integral part by X\/d<\/em>+O(1), this is clearly asymptotic to CX <\/em>with<\/p>\n

$latex C=\\sum_{d\\geq 1}{\\mu(d)^2d^{-2}}$<\/p>\n

which is an absolutely convergent series. As an Euler product it is<\/p>\n

$latex \\prod_{p}{(1+p^{-2})}=\\prod_{p}{\\frac{1-p^{-4}}{1-p^{-2}}}=\\frac{\\zeta(2)}{\\zeta(4)}$<\/p>\n

as desired.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here is a striking example of the unfortunate (but probably unavoidable) dispersion of mathematics: at the end of the abstract of V. Arnold’s talk at the recent conference in honor of A. Douady, one can read: The Cesaro mean values K\u2560\u00e9 of the numbers K(n) tend, as n tends to \u00d4\u00ea\u00d7, to a finite limit … Continue reading An exercise<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1086,2],"tags":[],"class_list":["post-57","post","type-post","status-publish","format-standard","hentry","category-exercise","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/57","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=57"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/57\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=57"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=57"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=57"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}