{"id":31,"date":"2018-01-08T16:29:33","date_gmt":"2018-01-08T16:29:33","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=31"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"zeta-dzeta","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/zeta-dzeta\/","title":{"rendered":"Riemann zeta Function $$\\zeta$$ and its Derivative $$\\zeta&#8217;$$"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-32 size-large\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/zeta_0-60_with_roots-1024x205.png\" alt=\"Riemann zeta Function $$\\zeta$$ and its Derivative $$\\zeta'$$\" width=\"1024\" height=\"205\" srcset=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/zeta_0-60_with_roots-1024x205.png 1024w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/zeta_0-60_with_roots-300x60.png 300w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/zeta_0-60_with_roots-768x154.png 768w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/zeta_0-60_with_roots.png 2000w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>The absolute value of the Riemann zeta function $$\\zeta(\\sigma+it)$$ for $$0 \\le \\sigma \\le 8$$ and $$0.1 \\le t \\le 60$$ over the complex plane. The white dots in the valleys are the non trivial zeros of the function. The Riemann hypothesis claims that the real part of all these zeros is $$1\/2$$.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-33 size-large\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/d_zeta_0-60_with_roots-1024x205.png\" alt=\"Riemann zeta Function $$\\zeta$$ and its Derivative $$\\zeta'$$\" width=\"1024\" height=\"205\" srcset=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/d_zeta_0-60_with_roots-1024x205.png 1024w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/d_zeta_0-60_with_roots-300x60.png 300w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/d_zeta_0-60_with_roots-768x154.png 768w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/d_zeta_0-60_with_roots.png 2000w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>The absolute value of the derivative $$\\zeta'(\\sigma+it)$$ of the Riemann zeta function for $$0 \\le \\sigma \\le 8$$ and $$0.1 \\le t \\le 60$$. The white dots in the valleys are zeros of the function. Both functions were plotted with the computer algebra system <a href=\"http:\/\/www.sagemath.org\">Sage<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The absolute value of the Riemann zeta function $$\\zeta(\\sigma+it)$$ for $$0 \\le \\sigma \\le 8$$ and $$0.1 \\le t \\le 60$$ over&#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-31","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\/31","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=31"}],"version-history":[{"count":4,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/31\/revisions"}],"predecessor-version":[{"id":246,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/31\/revisions\/246"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=31"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=31"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=31"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}