{"id":25,"date":"2018-01-08T16:24:07","date_gmt":"2018-01-08T16:24:07","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=25"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"zeta-chains","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/zeta-chains\/","title":{"rendered":"Zeros of the derivatives of the Riemann zeta function"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-26 size-full\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/figchains.png\" alt=\"Zeros of the derivatives of the Riemann zeta function\" width=\"597\" height=\"609\" srcset=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/figchains.png 597w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/figchains-294x300.png 294w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/p>\n<p>The plot shows zeros k of the derivatives $$\\zeta^{(k)}(\\sigma+it)$$ of the Riemann Zeta functionon the complex plane. In 1965 Spira had already noticed that the zeros of $$\\zeta'(s)$$ and $$\\zeta&#8221;(s)$$ seem to come in pairs, where the zero of $$\\zeta&#8221;(s)$$ is always located to the right of the zero of $$\\zeta'(s)$$. With the help of extensive computations, Skorokhodov (2003) observed this behavior for higher derivatives as well. For large $$k$$ and $$\\sigma$$ this phenomenon is proven in <a href=\"https:\/\/math-sites.uncg.edu\/sites\/pauli\/publications\/binder-pauli-saidak_new-zero-free-regions.pdf\"> New Zero-Free Regions for the Derivatives of the Riemann Zeta Function <\/a> by <a href=\"http:\/\/www.math.TU-Berlin.de\/%7Etbinder\">Thomas Binder<\/a>, <a href=\"https:\/\/math-sites.uncg.edu\/directory\/pauli\/\">Sebastian Pauli<\/a>, and <a href=\"https:\/\/math-sites.uncg.edu\/directory\/saidak\/\">Filip Saidak<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The plot shows zeros k of the derivatives $$\\zeta^{(k)}(\\sigma+it)$$ of the Riemann Zeta functionon the complex plane. In 1965 Spira had already&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-25","post","type-post","status-publish","format-standard","hentry","category-number-theory"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/25","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/comments?post=25"}],"version-history":[{"count":4,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/25\/revisions"}],"predecessor-version":[{"id":248,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/25\/revisions\/248"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=25"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=25"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=25"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}