{"id":413,"date":"2020-07-24T19:03:15","date_gmt":"2020-07-24T19:03:15","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=413"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"fractional-derivatives-of-the-riemann-zeta-function","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/fractional-derivatives-of-the-riemann-zeta-function\/","title":{"rendered":"Fractional Derivatives of the Riemann zeta function"},"content":{"rendered":"<p>\nThe video below shows the \u03b1-th Gr\u00fcnwald-Letnikov fractional derivative of the Riemann Zeta Function \u03b6(s) for \u03b1 between 0 and 10 on -20 \u2264 R(s) \u2264 20 and \u22123 \u2264 I(s) \u2264 27.\n<\/p>\n<p>\nThe hue represents the argument with red representing the positive real direction and cyan the negative real direction, as shown on the right. Brightness represents absolute value, with 0 represented by black and with white representing infinity.   The plot on the right show the identity function.\n<\/p>\n<p>\nPlots were generated with <a href=\"https:\/\/www.sagemath.org\">SageMath<\/a>.  For more see <a href=\"https:\/\/math-sites.uncg.edu\/sites\/pauli\/publications\/Caparatta-Pauli-Saidak_Fractional-derivatives-of-polynomials.pdf\">Zeros of Fractional Derivatives of Polynomials <\/a> by Torre Caparatta, Sebastian Pauli and Filip Saidak.  The video was complied with <a href=\"https:\/\/ffmpeg.org\">ffmpeg<\/a>.\n<\/p>\n<p><iframe loading=\"lazy\" title=\"Fractional derivatives of the Riemann Zeta Function on -20 \u2264 \u03c3 \u2264 20 and \u22123 \u2264 t \u2264 27\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/mdL0DU821Uo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The video below shows the \u03b1-th Gr\u00fcnwald-Letnikov fractional derivative of the Riemann Zeta Function \u03b6(s) for \u03b1 between 0 and 10 on&#8230;<\/p>\n","protected":false},"author":12,"featured_media":416,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-413","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-number-theory"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/413","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\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/comments?post=413"}],"version-history":[{"count":24,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/413\/revisions"}],"predecessor-version":[{"id":487,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/413\/revisions\/487"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media\/416"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=413"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=413"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=413"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}