30:T7b5,plot = cplot({ zoom: false, domain: [ [-25, 25], [-30, 30] ], base: 10, f: ` // sqrt(TAU) #define stau 2.506628274631 vec2 gamma(vec2 z) { const int N = 8; float c = stau; vec2 s = vec2(c,0); float f = 1.0; for(int k=1;k 1`}. Using analytic continuation, he extended it to the entire complex plane except at ${tex`s = 1`}. Riemann then began studying the zeros of the zeta function - the values of ${tex`s`} for which ${tex`\zeta(s) = 0`} Some of these are easy to describe. The trivial zeros occur at the negative even integers, ${tex`s = -2,-4,-6,...`}, and arise naturally from the functional equation of the zeta function. The remaining zeros are called the nontrivial zeros. They all lie inside the critical strip ${tex`0<\operatorname{Re}(s)<1`}. While investigating these nontrivial zeros, Riemann discovered a remarkable connection between the zeros and the distribution of prime numbers. More precisely, their locations control the size of the error in approximating the prime-counting function. Riemann observed that every nontrivial zero known to him lay on the vertical line ${tex`\operatorname{Re}(s)=\dfrac12`}, known as the critical line. This led him to propose the Riemann Hypothesis: > Every nontrivial zero of the Riemann zeta function lies on the critical line ${tex`\operatorname{Re}(s)=\dfrac12`}. Despite enormous computational evidence and over a century and a half of effort, no one has proved or disproved this conjecture. It remains one of the most famous unsolved problems in mathematics. ---32:T61b,zeta = { function cadd([a, b], [c, d]) { return [a + c, b + d]; } function csub([a, b], [c, d]) { return [a - c, b - d]; } function cmul([a, b], [c, d]) { return [a * c - b * d, a * d + b * c]; } function cscale([a, b], k) { return [a * k, b * k]; } function cdiv([a, b], [c, d]) { const den = c * c + d * d; return [(a * c + b * d) / den, (b * c - a * d) / den]; } function cexp([a, b]) { const ea = Math.exp(a); return [ea * Math.cos(b), ea * Math.sin(b)]; } function rpow(x, [a, b]) { const ln = Math.log(x); const mag = Math.exp(a * ln); return [mag * Math.cos(b * ln), mag * Math.sin(b * ln)]; } function cabs([a, b]) { return Math.hypot(a, b); } function zeta(s, maxN = 80, eps = 1e-13) { const [re, im] = s; if (Math.abs(re - 1) < 1e-15 && Math.abs(im) < 1e-15) { return [Infinity, Infinity]; } const oneMinusS = [1 - re, -im]; const twoPowOneMinusS = cexp(cscale(oneMinusS, Math.LN2)); const denom = csub([1, 0], twoPowOneMinusS); let sum = [0, 0]; for (let n = 0; n < maxN; n++) { let inner = [0, 0]; let coeff = 1; let sign = 1; for (let k = 0; k <= n; k++) { const term = cscale(rpow(k + 1, [-re, -im]), sign * coeff); inner = cadd(inner, term); sign = -sign; coeff *= (n - k) / (k + 1); } const outer = cscale(inner, 1 / Math.pow(2, n + 1)); sum = cadd(sum, outer); if (cabs(outer) < eps) break; } return cdiv(sum, denom); } return zeta; }33:[] 2e:{"title":"Riemann Zeta Function","cells":[{"id":0,"value":"# Riemann Zeta Function","pinned":false,"hidden":false,"mode":"md"},{"id":611,"value":"The Riemann zeta function is one of the most important objects in mathematics. \nTo understand where this function comes from, we first need to go back more than a century before Riemann.","pinned":false,"hidden":false,"mode":"md"},{"id":179,"value":"\\zeta(s) = \\sum_{n=1}^{\\infty} \\frac{1}{n^s}","pinned":false,"hidden":false,"mode":"tex"},{"id":346,"value":"tex`\\zeta(${complex(s)}) \\approx ${complex(zeta(s))}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":312,"value":"viewof s = Inputs.form([\n Inputs.number({ label: \"Real\", value: 0.5, step: 0.001 }),\n Inputs.number({ label: \"Imaginary\", value: 14.134724 })\n])","pinned":false,"hidden":false,"mode":"ojs"},{"id":491,"value":"$30","pinned":false,"hidden":false,"mode":"ojs"},{"id":351,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":642,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":629,"value":"complex = ([re, im]) => {\n if (Number.isNaN(re) || Number.isNaN(im)) {\n return \"\\\\mathrm{undefined}\";\n }\n if (!Number.isFinite(re) || !Number.isFinite(im)) {\n return \"\\\\infty\";\n }\n const clean = (x) => {\n if (!Number.isFinite(x)) return String(x);\n const ax = Math.abs(x);\n if (ax >= 1e6 || (ax > 0 && ax < 1e-4)) {\n const [m, e] = x.toExponential(3).split(\"e\");\n return `${m}\\\\times10^{${Number(e)}}`;\n }\n return Number(x.toPrecision(6)).toString();\n };\n if (Math.abs(re) < 1e-6) re = 0;\n if (Math.abs(im) < 1e-6) im = 0;\n const r = clean(re);\n const i = clean(Math.abs(im));\n if (im === 0) return r;\n if (re === 0) return `${im < 0 ? \"-\" : \"\"}${i}i`;\n\n return `${r}${im > 0 ? \"+\" : \"-\"}${i}i`;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":494,"value":"import { cplot } from \"@nxrix/domain-coloring\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W33","stdlib":"1","titling":"auto"}