{"id":16,"date":"2008-03-08T10:35:12","date_gmt":"2008-03-08T09:35:12","guid":{"rendered":"https:\/\/wpethzprd.ethz.ch\/kowalski\/2008\/03\/08\/averages-of-singular-series-or-when-poisson-is-everywhere\/"},"modified":"2024-02-20T13:41:12","modified_gmt":"2024-02-20T11:41:12","slug":"averages-of-singular-series-or-when-poisson-is-everywhere","status":"publish","type":"post","link":"https:\/\/blogs.ethz.ch\/kowalski\/2008\/03\/08\/averages-of-singular-series-or-when-poisson-is-everywhere\/","title":{"rendered":"Averages of singular series, or: when Poisson is everywhere"},"content":{"rendered":"<p> Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text&#8230;<\/p>\n<p>I have recently posted on my web page a <a href=\"http:\/\/www.math.ethz.ch\/~kowalski\/singular-series-distribution.pdf\">preprint<\/a> concerning some averages of &#8220;singular series&#8221; (another example of pretty bad mathematical terminology&#8230;) arising in the prime <em>k<\/em>-tuple conjecture, and its generalization the Bateman-Horn conjecture. The reason for looking at this is a result of Gallagher which is important in the original version of the proof by <a href=\"http:\/\/front.math.ucdavis.edu\/0512.5436\">Goldston-Pintz-Yildirim<\/a> that there are infinitely many primes <em>p<\/em> for which the gap <em>q-p<\/em> between <em>p<\/em> and the next prime <em>q<\/em> is smaller than <em>\u256c\u00c1<\/em> times the average gap, for arbitrary small <em>\u256c\u00c1&gt;0<\/em>.<\/p>\n<p>This result refers to the behavior, on average over <em><strong>h<\/strong>=(h_1,&#8230;,h_k)<\/em>, of the constant <em>S(<strong>h<\/strong>)<\/em> which is supposed to be the leading coefficient in the conjecture<\/p>\n<p align=\"center\"> |{<em>n<\/em>&lt;<em>X<\/em> | <em>n<\/em>+<em>h_i<\/em> is prime for <em>i<\/em>=1,&#8230;,<em>k<\/em>}|~<em>S(<strong>h<\/strong>)<\/em> <em>X<\/em>(log <em>X<\/em>)<sup><em>-k<\/em><\/sup><\/p>\n<p> Gallagher showed that the average value of <em>S(<strong>h<\/strong>)<\/em> is equal to <em>1<\/em>, and I&#8217;ve extended this in two ways&#8230;<\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2008\/03\/singular-series1.png\" title=\"[LaTeX 1]\"><img decoding=\"async\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2008\/03\/singular-series1.png\" alt=\"[LaTeX 1]\" \/><\/a><\/p>\n<p><a href=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2008\/03\/singular-series2.png\" title=\"[LaTex 2]\"><img decoding=\"async\" src=\"https:\/\/blogs.ethz.ch\/kowalski\/files\/2008\/03\/singular-series2.png\" alt=\"[LaTex 2]\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is another post where the mediocre mathematical abilities of HTML will require inserting images with some TeX-produced text&#8230; I have recently posted on my web page a preprint concerning some averages of &#8220;singular series&#8221; (another example of pretty bad mathematical terminology&#8230;) arising in the prime k-tuple conjecture, and its generalization the Bateman-Horn conjecture. The &hellip; <a href=\"https:\/\/blogs.ethz.ch\/kowalski\/2008\/03\/08\/averages-of-singular-series-or-when-poisson-is-everywhere\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Averages of singular series, or: when Poisson is everywhere<\/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-16","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/16","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=16"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/16\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=16"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=16"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=16"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}