{"id":67,"date":"2023-07-05T12:22:36","date_gmt":"2023-07-05T12:22:36","guid":{"rendered":"https:\/\/jonhough.com\/blog\/?p=67"},"modified":"2023-07-05T12:23:23","modified_gmt":"2023-07-05T12:23:23","slug":"plotting-the-riemann-siegel-formula-with-j","status":"publish","type":"post","link":"https:\/\/jonhough.com\/blog\/2023\/07\/05\/plotting-the-riemann-siegel-formula-with-j\/","title":{"rendered":"Plotting the Riemann-Siegel Formula with J"},"content":{"rendered":"\n<h3 class=\"wp-block-heading\">Plotting Riemann-Siegel Formula with J<\/h3>\n\n\n\n<p>The Riemann-Siegel formula, Z(t) has been used to find non-trivial zeros of Riemann&#8217;s zeta function.<br><a href=\"https:\/\/en.wikipedia.org\/wiki\/Riemann%E2%80%93Siegel_formula\">wikipedia<\/a><\/p>\n\n\n\n<p>The following source code gives a numerical calculation of <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/jonhough.com\/blog\/wp-content\/ql-cache\/quicklatex.com-1d447861c0cced8a2afbabb9a597978d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#90;&#40;&#116;&#41;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#50;&#32;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#78;&#125;&#110;&#8722;&#49;&#50;&#32;&#92;&#99;&#111;&#115;&#91;&#92;&#116;&#104;&#101;&#116;&#97;&#40;&#116;&#41;&#8722;&#116;&#32;&#92;&#108;&#111;&#103;&#32;&#110;&#93;&#43;&#82;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"295\" style=\"vertical-align: -5px;\"\/><script src=\"https:\/\/jonhough.com\/blog\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/jonhough.com\/blog\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<p>where<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/jonhough.com\/blog\/wp-content\/ql-cache\/quicklatex.com-7794f95d318c908a6b1709f68c5dc925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#40;&#116;&#41;&#32;&#61;&#32;&#92;&#73;&#109;&#32;&#92;&#108;&#111;&#103;&#32;&#92;&#71;&#97;&#109;&#109;&#97;&#32;&#40;&#49;&#32;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#105;&#116;&#125;&#123;&#50;&#125;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#41;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#125;&#92;&#108;&#111;&#103;&#92;&#112;&#105;&#32;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"272\" style=\"vertical-align: -6px;\"\/><\/div>\n\n\n\n<p>&nbsp;This can be simplified to an easily calculatable approximation given by:<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/jonhough.com\/blog\/wp-content\/ql-cache\/quicklatex.com-047ea662571819c4512db687103cfb41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#40;&#116;&#41;&#32;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#125;&#92;&#108;&#111;&#103;&#32;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#92;&#32;&#112;&#105;&#125;&#41;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#125;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#112;&#105;&#125;&#123;&#56;&#125;&#32;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#56;&#116;&#32;&#125;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#53;&#55;&#54;&#48;&#116;&#94;&#123;&#51;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"311\" style=\"vertical-align: -9px;\"\/><\/div>\n\n\n\n<p>The final term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/jonhough.com\/blog\/wp-content\/ql-cache\/quicklatex.com-fb7a7b1837b81b4e9bc9bddfacb5e884_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"18\" style=\"vertical-align: -4px;\"\/> in the original sum, is the remainder term, which is complicated to derive but \u00a0can be reduced to a sum,<\/p>\n\n\n\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/jonhough.com\/blog\/wp-content\/ql-cache\/quicklatex.com-37d240ccf0346aa455fb24822a9eb936_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#32;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#40;&#45;&#49;&#41;&#94;&#123;&#78;&#45;&#49;&#125;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#41;&#94;&#123;&#45;&#49;&#47;&#52;&#125;&#91;&#67;&#95;&#123;&#48;&#125;&#32;&#43;&#32;&#67;&#95;&#123;&#49;&#125;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#41;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#32;&#43;&#67;&#95;&#123;&#50;&#125;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#41;&#94;&#123;&#45;&#49;&#125;&#43;&#67;&#95;&#123;&#51;&#125;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#41;&#94;&#123;&#45;&#51;&#47;&#50;&#125;&#32;&#43;&#32;&#67;&#95;&#123;&#52;&#125;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#116;&#125;&#123;&#50;&#32;&#92;&#112;&#105;&#125;&#41;&#94;&#123;&#45;&#50;&#125;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"605\" style=\"vertical-align: -6px;\"\/><\/p>\n\n\n\n<p>We will use this approximation in the calculation.<\/p>\n\n\n\n<p><\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<pre class=\"wp-block-code\"><code>\nNB. Calculation of Z(x), estimate of zeta(x) for x = a +ib where b = 0.5  \nNB. Riemann-Siegel formula. (from Riemann Zeta Function, H.M Edwards)      \n\nload 'plot'  \nload 'numeric trig'  \n    \nC0 =: 0.38268343236508977173, 0.43724046807752044936, 0.13237657548034352333, _0.01360502604767418865, _0.01356762197010358088, _0.00162372532314446528, 0.00029705353733379691,0.00007943300879521469, 0.00000046556124614504, _0.00000143272516309551, _0.00000010354847112314, 0.00000001235792708384, 0.00000000178810838577, _0.00000000003391414393, _0.00000000001632663392  \n\nC1 =: 0.02682510262837535, _0.01378477342635185, _0.03849125048223508, _0.00987106629906208, 0.00331075976085840, 0.00146478085779542, 0.00001320794062488,_0.00005922748701847, _0.00000598024258537, 0.00000096413224562, 0.00000018334733722\n  \nC2 =: 0.005188542830293, 0.000309465838807, _0.011335941078299, 0.002233045741958,0.005196637408862, 0.000343991440762, _0.000591064842747, _0.000102299725479, 0.000020888392217,0.000005927665493, _0.000000164238384, _0.000000151611998  \n\nC3 =: 0.0013397160907, _0.0037442151364, 0.0013303178920, 0.0022654660765,_0.0009548499998, _0.0006010038459, 0.0001012885828, 0.0000686573345, _0.0000005985366,_0.0000033316599, _0.0000002191929, 0.0000000789089, 0.0000000094147 \n \nC4 =: 0.00046483389, _0.00100566074, 0.00024044856, 0.00102830861,_0.00076578609, _0.00020365286, 0.00023212290, 0.00003260215, _0.00002557905, _0.00000410746,0.00000117812, 0.00000024456  \n    \npi2 =: +: o. 1  \n    \ntheta =: 3 : 0  \nassert. y &gt; 0  \np1 =. ( * -:@:^.@:(%&amp;pi2)) y  \np2 =. --: y  \np3 =. -(o. 1) % 8  \np4 =. 1 % (48 * y)  \np5 =. 7 % (5760 * y ^ 3)  \np1 + p2 + p3 + p4 + p5  \n)  \n    \nsummation =: 3 : 0 \"1  \n   \nv =. &gt;0{y  \nm =. &gt;1{y  \nthetav =. theta v  \nms =. &gt;: i. m  \ncos =. 2&amp;o.  \ns =. +:@:(^&amp;_0.5) * cos@:(thetav&amp;-@:(v&amp;*@:^.))  \n+\/ s ms  \n)  \n    \nremainderterms =: 3 : 0  \nim =. y  \nv =. %: pi2 %~ y  \nN =. &lt;. v  \np =. v - N  \nq =. 1 - 2 * p  \nfinal =. 0  \n   \nmv0 =. +\/ (({&amp;C0) * (q&amp;^@:+:)) i. # C0  \nfinal =. mv0  \n  \nmv1 =. +\/ (({&amp;C1) * (q&amp;^@:+:@:&gt;:)) i. # C1  \nmv1 =. mv1 * ((im % pi2) ^ _0.5)  \nfinal =. final + mv1  \n   \nmv2 =. +\/ (({&amp;C2) * (q&amp;^@:+:)) i. # C2  \nmv2 =. mv2 * ((im % pi2) ^ _1)  \nfinal =. final + mv2  \n    \nmv3 =. +\/ (({&amp;C3) * (q&amp;^@:+:@:&gt;:)) i. # C3  \nfinal =. final + mv3  \n    \nmv4 =. +\/ (({&amp;C4) * (q&amp;^@:+:)) i. # C4  \nfinal =. final + mv4  \n   \n    \nfinal * (_1 ^ &lt;: N) * ((im % pi2) ^ _0.25)  \n)  \n   \n    \nz =: 3 : 0 \"0  \nim =. y  \nv =. %: pi2 %~ y  \nN =. &lt;. v  \np =. v - N  \nv1 =. summation im ; N  \nv2 =. remainderterms im  \nv1 + v2  \n)     \n   \nx =: steps 0.1 100 3500  \nplot x; z x  \n<\/code><\/pre>\n<\/div><\/div>\n\n\n\n<h2 class=\"wp-block-heading\">Graphical plot<\/h2>\n\n\n\n<p>The lines:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code> x =: steps 0.1 100 3500  \n plot x; z x  \n<\/code><\/pre>\n\n\n\n<p>create a plot of the function from 0.1 to 100 in 3500 steps, as shown below:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/4.bp.blogspot.com\/-wmdRXe3e-jg\/WCrZ8vC6zKI\/AAAAAAAAAOk\/TLYoETXhfu4tT8d2JoiUEoepFMPn1YLCgCLcB\/s1600\/zeta1.png\"><img decoding=\"async\" src=\"https:\/\/4.bp.blogspot.com\/-wmdRXe3e-jg\/WCrZ8vC6zKI\/AAAAAAAAAOk\/TLYoETXhfu4tT8d2JoiUEoepFMPn1YLCgCLcB\/s320\/zeta1.png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<p>You can see the zeros of the&nbsp;<em>Zeta Function&nbsp;<\/em>from this graph. The first zero is at about&nbsp;<em>x = 14.29<\/em>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Plotting Zeta function on the complex plane<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code> zeta3d =: 3 : 0  \n if. (0{+.y) &gt; 0.5 do.  \n I =. &gt;: i. 100  \n +\/I^-y  \n elseif. (0{+.y) &lt; 0.5 do.  \n z =. zeta3d 1-y  \n g =. ! 1-y  \n s =. 1 o. -:(o.1)*y  \n c =. ((o.1)^(y-1))*2^y  \n r=.z*g*s*c  \n if. 40 &lt; |r do.  \n r =. 40  \n end.  \n r  \n elseif. 1 do.  \n 0  \n end.  \n )  \n   \n absz =: |@:zeta3d  \n realz=: 0&amp;{@:+.@:zeta3d  \n imz=: 1&amp;{@:+.@:zeta3d  \n D =.steps _15 15 40  \n 'surface; viewpoint 1 _1 2' plot imz\"0 j.\/~ D NB. use realz for real graph.\n<\/code><\/pre>\n\n\n\n<p>Here is a plot of the imaginary parts:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/3.bp.blogspot.com\/-PPMJY6wx8g0\/WD6AAWornpI\/AAAAAAAAAO4\/lq6UJv54kXcG_usq2KgSb0SwnJ4Al7ZEACLcB\/s1600\/imaginary.png\"><img decoding=\"async\" src=\"https:\/\/3.bp.blogspot.com\/-PPMJY6wx8g0\/WD6AAWornpI\/AAAAAAAAAO4\/lq6UJv54kXcG_usq2KgSb0SwnJ4Al7ZEACLcB\/s320\/imaginary.png\" alt=\"\"\/><\/a><\/figure>\n\n\n\n<p>&#8230;.and the real parts.<\/p>\n\n\n\n<figure class=\"wp-block-image\"><a href=\"https:\/\/2.bp.blogspot.com\/-zWKioREX3uY\/WD6AARFJl8I\/AAAAAAAAAO8\/Ww-P43ZXZ1kbvXVO-kjxd5lh8UMO-FsKACLcB\/s1600\/real.png\"><img decoding=\"async\" src=\"https:\/\/2.bp.blogspot.com\/-zWKioREX3uY\/WD6AARFJl8I\/AAAAAAAAAO8\/Ww-P43ZXZ1kbvXVO-kjxd5lh8UMO-FsKACLcB\/s320\/real.png\" alt=\"\"\/><\/a><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Plotting Riemann-Siegel Formula with J The Riemann-Siegel formula, Z(t) has been used to find non-trivial zeros of Riemann&#8217;s zeta function.wikipedia The following source code gives a numerical calculation of where &nbsp;This can be simplified to an easily calculatable appro<a class=\"read-more\" href=\"#\"><span class=\"read-more-text\">Read More <\/span><i class=\"fa fa-angle-double-down\"><\/i><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","om_disable_all_campaigns":false,"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","footnotes":""},"categories":[21,3,1],"tags":[25],"class_list":["post-67","post","type-post","status-publish","format-standard","hentry","category-j","category-tech","category-uncategorized","tag-j-zeta-math"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - 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