{"id":1591,"date":"2010-02-15T20:25:51","date_gmt":"2010-02-15T18:25:51","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/?p=1591"},"modified":"2024-02-20T13:40:54","modified_gmt":"2024-02-20T11:40:54","slug":"addendum","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2010\/02\/15\/addendum\/","title":{"rendered":"Addendum"},"content":{"rendered":"<p>Here is a quick update to <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2010\/02\/09\/cauchy-joins-the-club\/\">the last post<\/a> about (restricted) mod-Cauchy convergence; I&#8217;ve investigated numerically the behavior of the renormalized averages<\/p>\n<p>$latex \\mathbf{E}_N(e^{its(d,c)}) \\times \\exp(\\gamma_N |t|)$<\/p>\n<p>(see the post for the notation) for some values of <em>t<\/em>, to see if the limitation to<\/p>\n<p>$latex |t|&lt;2\\pi$<\/p>\n<p>in Vardi&#8217;s result could be a mere artifact of the method.  Here are some graphs representing these empirical averages for<\/p>\n<p>$latex N\\leq 5000$<\/p>\n<p>(click to see the full pictures):<\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-halfpi.jpg\"><img decoding=\"async\" class=\"aligncenter size-medium wp-image-1592\" style=\"border: 1px solid black\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-halfpi.jpg\" alt=\"\" width=\"500\" \/><\/a><\/p>\n<p>$latex t=\\frac{\\pi}{2}$<\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-pi.jpg\"><img decoding=\"async\" class=\"aligncenter size-medium wp-image-1597\" style=\"border: 1px solid black\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-pi.jpg\" alt=\"\" width=\"500\" \/><\/a><\/p>\n<p>$latex t=\\pi$<\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-2pi.jpg\"><img decoding=\"async\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-2pi.jpg\" alt=\"\" style=\"border: 1px solid black\" width=\"500\" class=\"aligncenter size-medium wp-image-1598\" \/><\/a><\/p>\n<p>$latex t=2\\pi$<\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-4pi.jpg\"><img decoding=\"async\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2010\/02\/dedekind-4pi.jpg\" alt=\"\" style=\"border: 1px solid black\" width=\"500\" class=\"aligncenter size-medium wp-image-1599\" \/><\/a><\/p>\n<p>$latex t=4\\pi$<\/p>\n<p>In particular, note that in the last picture, the vertical scale runs from -300 to 300, more or less, compared with oscillations between 0.5 and 1.5 in the third. So it seems pretty convincing evidence that the limit as <i>N<\/i> goes to infinity does not exist when <i>t<\/i> is large.<\/p>\n<p>(<u>Note<\/u>: the empirical average for <i>N=5000<\/i> involves about 8,000,000 Dedekind sums).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a quick update to the last post about (restricted) mod-Cauchy convergence; I&#8217;ve investigated numerically the behavior of the renormalized averages $latex \\mathbf{E}_N(e^{its(d,c)}) \\times \\exp(\\gamma_N |t|)$ (see the post for the notation) for some values of t, to see if the limitation to $latex |t|&lt;2\\pi$ in Vardi&#8217;s result could be a mere artifact of &hellip; <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2010\/02\/15\/addendum\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Addendum<\/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":[2],"tags":[],"class_list":["post-1591","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/1591","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=1591"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/1591\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=1591"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=1591"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=1591"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}