{"id":4118,"date":"2016-04-23T20:37:40","date_gmt":"2016-04-23T20:37:40","guid":{"rendered":"http:\/\/theoryofeverything.org\/theToE\/?p=4118"},"modified":"2016-05-03T18:45:31","modified_gmt":"2016-05-03T18:45:31","slug":"interactive-reimann-zeta-function-zeros-demonstration","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2016\/04\/23\/interactive-reimann-zeta-function-zeros-demonstration\/","title":{"rendered":"Interactive Reimann Zeta Function Zeros Demonstration"},"content":{"rendered":"<p>This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including <a href=\"https:\/\/en.wikipedia.org\/wiki\/Riemann_hypothesis#Gram_points\">Gram points<\/a>) along the critical line Zeta(1\/2+it) for real values of t running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t.  Gram&#8217;s law states that the curve usually crosses the real axis once between zeros.<\/p>\n<p>A downloadable copy is available <a href=\"\/TOE\/JGM\/ZetaZeros-local-art-50.cdf\">ZetaZeros-local-art-50.cdf<\/a> Latest:  05\/23\/2016.<\/p>\n<p>Note: The interactive CDF plug-in as required below does not currently work on Chrome browsers.<\/p>\n<p><script type=\"text\/javascript\" src=\"http:\/\/www.wolfram.com\/cdf-player\/plugin\/v2.1\/cdfplugin.js\"><\/script><script type=\"text\/javascript\">\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\nvar cdf = new cdfplugin();\ncdf.embed('http:\/\/theoryofeverything.org\/TOE\/JGM\/ZetaZeros-local-art-50.cdf',1000,1000);\n\/\/ ]]><\/script><\/p>\n<p>A Snapshot picture for those w\/o Wolfram CDF interactivity:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/RiemannZeta_Zeros.svg_-1.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/RiemannZeta_Zeros.svg_-1.png\" alt=\"RiemannZeta_Zeros.svg\" width=\"2000\" height=\"1744\" class=\"aligncenter size-full wp-image-4148\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/RiemannZeta_Zeros.svg_-1.png 2000w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/RiemannZeta_Zeros.svg_-1-500x436.png 500w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/RiemannZeta_Zeros.svg_-1-1024x893.png 1024w\" sizes=\"auto, (max-width: 2000px) 100vw, 2000px\" \/><\/a><\/p>\n<p>Selectable example code snippet:<br \/>\n[wlcode]Show[ListPlot[Style[pts2, Red], PlotRange -> {{-2, 4}, {-3, 3}},<br \/>\n  AspectRatio -> 1, ImageSize -> imageSize,<br \/>\n  AxesStyle -><br \/>\n   Directive[Thick, If[artPrint &#038;&#038; ! localize, Large, Medium]],<br \/>\n Graphics[{PointSize@.01,<br \/>\n   tttxt := If[artPrint &#038;&#038; ! localize, tttxt1, ttxt0];<br \/>\n   If[ttxt0 = ToString[# &#8211; 1];<br \/>\n      Abs@zeroY[[#, 1]] < 10 chop,\n      (* Magenta Critical Line Zeta Zeros *)\n      tttxt1 = \n       Column[{ToString[# - 1], \"t=\" <> ToString@zeroY[[#, 4]]},<br \/>\n        Center];<br \/>\n      ttLoc =<br \/>\n       zeroY[[#, 4]] If[artPrint &#038;&#038; ! localize, 1, 2] imagesize\/40000;<br \/>\n      (* Flip Point Labels above\/below the X axis *)<br \/>\n      ttLoc1 = {-1, (-1)^Round[#\/2]} ttLoc\/Sqrt[2];<br \/>\n      {Magenta, Point@ttLoc1,<br \/>\n       Circle[zeroY[[#, ;; 2]], ttLoc, {1, 3} \\[Pi]\/2],<br \/>\n       Black,<br \/>\n       Tooltip[Text[<br \/>\n         Style[tttxt, If[artPrint &#038;&#038; ! localize, Large, Medium, Bold]],<br \/>\n         (* Shift the Labels off the Point *)<br \/>\n         ttLoc1 (1 &#8211; (-1)^Round[#\/2] .05)], tttxt1]},<br \/>\n      (* Orange Critical Line Imaginary zeros w\/Real>0  *)<br \/>\n      tttxt1 =<br \/>\n       Column[{ToString[# &#8211; 1], &#8220;x=&#8221; <> ToString@zeroY[[#, 1]],<br \/>\n         &#8220;t=&#8221; <> ToString@zeroY[[#, 4]]}, Center];<br \/>\n      {Tooltip[{Orange, Point@zeroY[[#, ;; 2]], Text[Style[<\/p>\n<p>           Column[If[EvenQ[Round[(# &#8211; 1)\/2]], Prepend,<br \/>\n              Append][{&#8220;\\[UpDownArrow]&#8221;}, tttxt], Center,<br \/>\n            Frame -> True],<br \/>\n           Black, If[artPrint &#038;&#038; ! localize, Large, Medium],<br \/>\n           Background -> White],<br \/>\n          (* Flip Point Labels above\/below the X axis *)<\/p>\n<p>          zeroY[[#, ;; 2]] + {0, (-1)^Round[(# &#8211; 1)\/2]} If[<br \/>\n             artPrint &#038;&#038; ! localize, 1,<br \/>\n             If[artPrint, 4\/1, 3]] imagesize\/5000]},<br \/>\n        Column[{ToString[# &#8211; 1], &#8220;x=&#8221; <> ToString@zeroY[[#, 1]],<br \/>\n          &#8220;t=&#8221; <> ToString@zeroY[[#, 4]]}, Center]]}] &#038; \/@<br \/>\n    Range@Length@zeroY,<br \/>\n   Magenta, Disk[{0, 0}, .03]}]][\/wlcode]<\/p>\n<p>More plots with various scaling functions and multi-color coding along with Tooltip on mouse-over. Bear in mind the last Smith Chart with a division by Abs@Zeta indicates where the increments go exponential near the 0.<br \/>\n<script type=\"text\/javascript\" src=\"http:\/\/www.wolfram.com\/cdf-player\/plugin\/v2.1\/cdfplugin.js\"><\/script><script type=\"text\/javascript\">\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\n\/\/ < ![CDATA[\nvar cdf = new cdfplugin();\ncdf.embed('http:\/\/theoryofeverything.org\/TOE\/JGM\/ZetaZeros.cdf',938,4200);\n\/\/ ]]><\/script><\/p>\n<p>A Snapshot picture for those w\/o Wolfram CDF interactivity:<br \/>\n<a href=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/ZetaZeros.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/ZetaZeros.png\" alt=\"ZetaZeros\" width=\"938\" height=\"4388\" class=\"aligncenter size-full wp-image-4188\" srcset=\"https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/ZetaZeros.png 938w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/ZetaZeros-107x500.png 107w, https:\/\/theoryofeverything.org\/theToE\/wp-content\/uploads\/2016\/04\/ZetaZeros-219x1024.png 219w\" sizes=\"auto, (max-width: 938px) 100vw, 938px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1\/2+it) for real values of t running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to &hellip; <a href=\"https:\/\/theoryofeverything.org\/theToE\/2016\/04\/23\/interactive-reimann-zeta-function-zeros-demonstration\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Interactive Reimann Zeta Function Zeros Demonstration<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"image","meta":{"footnotes":""},"categories":[3],"tags":[14,16],"class_list":["post-4118","post","type-post","status-publish","format-image","hentry","category-physics","tag-math","tag-physics","post_format-post-format-image"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.8 - aioseo.com -->\n\t<meta name=\"description\" content=\"This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1\/2+it) for real values of t running from 0 to 50. 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The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to"},"aioseo_meta_data":{"post_id":"4118","title":null,"description":null,"keywords":null,"keyphrases":null,"primary_term":null,"canonical_url":null,"og_title":null,"og_description":null,"og_object_type":"default","og_image_type":"default","og_image_url":null,"og_image_width":null,"og_image_height":null,"og_image_custom_url":null,"og_image_custom_fields":null,"og_video":null,"og_custom_url":null,"og_article_section":null,"og_article_tags":null,"twitter_use_og":false,"twitter_card":"default","twitter_image_type":"default","twitter_image_url":null,"twitter_image_custom_url":null,"twitter_image_custom_fields":null,"twitter_title":null,"twitter_description":null,"schema":{"blockGraphs":[],"customGraphs":[],"default":{"data":{"Article":[],"Course":[],"Dataset":[],"FAQPage":[],"Movie":[],"Person":[],"Product":[],"ProductReview":[],"Car":[],"Recipe":[],"Service":[],"SoftwareApplication":[],"WebPage":[]},"graphName":"","isEnabled":true},"graphs":[],"defaultGraph":"","defaultPostTypeGraph":""},"schema_type":"default","schema_type_options":null,"pillar_content":false,"robots_default":true,"robots_noindex":false,"robots_noarchive":false,"robots_nosnippet":false,"robots_nofollow":false,"robots_noimageindex":false,"robots_noodp":false,"robots_notranslate":false,"robots_max_snippet":null,"robots_max_videopreview":null,"robots_max_imagepreview":"large","priority":null,"frequency":null,"local_seo":null,"limit_modified_date":false,"created":"2022-04-06 04:18:11","updated":"2025-06-10 15:47:16"},"aioseo_breadcrumb":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\" title=\"Home\">Home<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\t<a href=\"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/\" title=\"Physics\">Physics<\/a>\n\t\t<\/span><span class=\"aioseo-breadcrumb-separator\">\u00bb<\/span><span class=\"aioseo-breadcrumb\">\n\t\t\tInteractive Reimann Zeta Function Zeros Demonstration\n\t\t<\/span><\/div>","aioseo_breadcrumb_json":[{"label":"Home","link":"https:\/\/theoryofeverything.org\/theToE"},{"label":"Physics","link":"https:\/\/theoryofeverything.org\/theToE\/topics\/physics\/"},{"label":"Interactive Reimann Zeta Function Zeros Demonstration","link":"https:\/\/theoryofeverything.org\/theToE\/2016\/04\/23\/interactive-reimann-zeta-function-zeros-demonstration\/"}],"_links":{"self":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/comments?post=4118"}],"version-history":[{"count":5,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4118\/revisions"}],"predecessor-version":[{"id":4196,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/posts\/4118\/revisions\/4196"}],"wp:attachment":[{"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/media?parent=4118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/categories?post=4118"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/theoryofeverything.org\/theToE\/wp-json\/wp\/v2\/tags?post=4118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}