{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Non trivial Zeros of Riemann Zeta knows where primes are\n", "Yasuaki Honda at gmail dot com\n", "\n", "[1] Conrey, The Riemann Hypothesis https://www.ams.org/notices/200303/fea-conrey-web.pdf\n", "\n", "[2] The LMFDB Collaboration, The L-functions and Modular Forms Database, \n", "home page of the Zeros of zeta(s),\n", "https://www.lmfdb.org/zeros/zeta/?limit=200&N=1, 2020 , [Online; accessed 10 October 2020]." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "\\[\\tag{${\\it \\%o}_{1}$}f\\left(x\\right)={\\it realpart}\\left(\\sum_{j=1}^{\\infty }{x^{\\rho_{j}}}\\right)\\]" ], "text/plain": [ " inf\n", " ==== rho\n", " \\ j\n", "(%o1) f(x) = realpart( > x )\n", " /\n", " ====\n", " j = 1" ], "text/x-maxima": [ "f(x) = 'realpart('sum(x^rho[j],j,1,inf))" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x)='realpart(sum(x^rho[j],j,1,inf));" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "\\[\\tag{${\\it \\%o}_{2}$}\\mathbf{done}\\]" ], "text/plain": [ "(%o2) done" ], "text/x-maxima": [ "done" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "/*\n", "The LMFDB Collaboration, The L-functions and Modular Forms Database, \n", "home page of the Zeros of zeta(s),\n", "https://www.lmfdb.org/zeros/zeta/?limit=200&N=1, 2020 , [Online; accessed 10 October 2020].\n", "*/\n", "kill(img_rho);\n", "img_rho:[\n", "14.1347251417346937904572519835625,\n", "21.0220396387715549926284795938969,\n", "25.0108575801456887632137909925628,\n", "30.4248761258595132103118975305840,\n", "32.9350615877391896906623689640747,\n", "37.5861781588256712572177634807053,\n", "40.9187190121474951873981269146334,\n", "43.3270732809149995194961221654068,\n", "48.0051508811671597279424727494277,\n", 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"353.4889004887188067836037686094353,\n", "356.0175749772649473179603619607666,\n", "357.1513022520396248096029282322904,\n", "357.9526851016322737551289189827774,\n", "359.7437549531144487992919859769902,\n", "361.2893616958046503902913112277154,\n", "363.3313305789738347473344495666148,\n", "364.7360241140889937162621021269403,\n", "366.2127102883313168610771451968066,\n", "367.9935754817403033261832980442079,\n", "368.9684380957343898915769012998921,\n", "370.0509192121060003396511630612665,\n", "373.0619283721128384491193964312852,\n", "373.8648739109085697447563627336596,\n", "375.8259127667393341079077141466931,\n", "376.3240922306680521171908196124275,\n", "378.4366802499654797240909659132034,\n", "379.8729753465323466510240596904463,\n", "381.4844686171865249196625224236574,\n", "383.4435294495364877043457554807227,\n", "384.9561168148636871037515842087935,\n", "385.8613008459742291805619593598032,\n", "387.2228902223879809759485147381099,\n", "388.8461283542322546008094203003735,\n", "391.4560835636380457705782281225541,\n", "392.2450833395190967490151841709930,\n", "393.4277438444340259366989529201288,\n", "395.5828700109937209708777113231417,\n", "396.3818542225921869319994544917305\n", "]$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "\\[\\tag{${\\it \\%o}_{4}$}f\\left(x\\right):=-{\\it ev}\\left({\\it realpart}\\left({\\it sum}\\left(x^{\\frac{1}{2}+i\\,{\\it img\\_rho}_{j}} , j , 1 , 200\\right)\\right) , {\\it numer}\\right)\\]" ], "text/plain": [ " 1/2 + %i img_rho\n", " j\n", "(%o4) f(x) := - ev(realpart(sum(x , j, 1, 200)), numer)" ], "text/x-maxima": [ "f(x):=-ev(realpart(sum(x^(1/2+%i*img_rho[j]),j,1,200)),numer)" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x):=-ev(realpart(sum(x^(1/2+%i*img_rho[j]),j,1,200)),numer);" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "pi1[n]:=if n<2 then 0 elseif primep(n) then pi1[n-1]+1 else pi1[n-1]$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "pi(x):=if integerp(x) and primep(x) then pi1[x]-1/2 else pi1[floor(x)]$" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/svg+xml": [ "\n", "\n", "\n", "Gnuplot\n", "Produced by GNUPLOT 5.2 patchlevel 8 \n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\t\n", "\t \n", "\t \n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\t\n", "\n", "\t\n", "\t\tx\n", "\t\n", "\n", "\n", "\n", "\tfun1\n", "\n", "\t\n", "\t\tfun1\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\tfun2\n", "\n", "\t\n", "\t\tfun2\n", "\t\n", "\n", "\n", "\t\n", "\t\n", "\n", "\t\t\n", "\t\t-200\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t-150\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t-100\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t-50\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 0\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 50\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 100\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 150\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 200\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 250\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 10\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 20\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 30\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 40\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 50\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 60\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 70\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 80\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 90\n", "\t\n", "\n", "\n", "\t\t\n", "\t\t 100\n", "\t\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\t\n", "\n", "\n", "\n", "\n", "\n" ], "text/plain": [ "/tmp/maxplot.svg" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "\\[\\tag{${\\it \\%o}_{7}$}\\left[ \\mbox{ /tmp/maxout614.gnuplot } , \\mbox{ /tmp/maxplot.svg } \\right] \\]" ], "text/plain": [ "(%o7) [/tmp/maxout614.gnuplot, /tmp/maxplot.svg]" ], "text/x-maxima": [ "[\"/tmp/maxout614.gnuplot\",\"/tmp/maxplot.svg\"]" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot2d([f(x),pi(x)*10],[x,1,100]);" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Maxima", "language": "maxima", "name": "maxima" }, "language_info": { "codemirror_mode": "maxima", "file_extension": ".mac", "mimetype": "text/x-maxima", "name": "maxima", "pygments_lexer": "maxima", "version": "5.44.0" } }, "nbformat": 4, "nbformat_minor": 4 }