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| "use strict"; | |
| class Complex { | |
| constructor(a, b) { | |
| this.re = a; | |
| this.im = b; | |
| } | |
| add(c) { | |
| return new Complex(this.re + c.re, this.im + c.im); | |
| } | |
| toString() { | |
| //return this.re + ' + i' + this.im; | |
| return this.abs() + ' e^i' + this.theta(); | |
| } | |
| theta() { | |
| return Math.atan(this.im / this.re); | |
| } | |
| abs() { | |
| return Math.sqrt(this.re * this.re + this.im * this.im); | |
| } | |
| } | |
| function zeta(a, b) { | |
| function val(n) { | |
| const na = Math.pow(n, -a); | |
| const logn = Math.log(n); | |
| return new Complex(na * Math.cos(b * logn), -na * Math.sin(b * logn)); | |
| } | |
| var z = val(1); | |
| //const max = 10000; | |
| const max = b / 2.25; | |
| for (var i = 2; i < max; i++) { | |
| var nz = val(i); | |
| z = z.add(nz); | |
| //console.log(i + "\tz=" + z.abs() + "\ttheta=" + z.theta() + "\tarc=" + (b * (Math.log(i) - Math.log(i - 1))) + '\tarc2=' + (b/i)); | |
| } | |
| return z; | |
| } | |
| function fmt(x) { | |
| var s = '' + Math.round(x * 1000); | |
| return s.substring(0, s.length - 3) + '.' + s.substring(s.length - 3); | |
| } | |
| function findZeros(min, len) { | |
| var lastAbs = 1; | |
| var inc = 1 / 1000; | |
| var up = false; | |
| var results = []; | |
| for (var ii = min; ii < min + len; ii += inc) { | |
| const z = zeta(0.5, ii); | |
| const abs = z.abs(); | |
| if (abs < lastAbs && up) { | |
| up = false; | |
| lastAbs = abs; | |
| } else if (abs > lastAbs && !up) { | |
| if (abs < 1 / Math.sqrt(ii)) { | |
| console.log(fmt(ii - inc) + '\t\t' + abs + '\t' + (1 / Math.sqrt(ii))); | |
| results.push(ii - inc); | |
| } | |
| lastAbs = abs; | |
| up = true; | |
| } else { | |
| up = abs > lastAbs; | |
| lastAbs = abs; | |
| } | |
| } | |
| return results; | |
| } | |
| function run(min, len) { | |
| const results = findZeros(min, len); | |
| const zeros = require('./zzeros').filter(x => x >= min && x < min + len); | |
| var i = 0; | |
| var j = 0; | |
| var matches = []; | |
| var bad = 0, missed = 0; | |
| while (i < results.length - 1) { | |
| var delta1 = Math.abs(results[i] - zeros[j]); | |
| var delta2 = Math.abs(results[i] - zeros[j + 1]); | |
| var delta3 = Math.abs(results[i + 1] - zeros[j]); | |
| if (delta1 < delta2 && delta1 < delta3) { | |
| matches.push([results[i], zeros[j]]); | |
| i++; | |
| j++; | |
| } else if (delta2 < delta3) { | |
| console.error(zeros[j] + ': missed'); | |
| missed++; | |
| j++; | |
| } else { | |
| console.error(fmt(results[i]) + ': bad'); | |
| bad++; | |
| i++; | |
| } | |
| } | |
| console.log(); | |
| console.log(bad + " bad results"); | |
| console.log(missed + " missed zeros"); | |
| const err = a => Math.abs(a[0] - a[1]) / a[1]; | |
| matches.sort((a, b) => err(a) - err(b)); | |
| const fmtPair = x => fmt(x[0]) + '\t' + x[1] + '\t' + err(x); | |
| console.log("\nbest matches:\n" + matches.slice(0, 20).map(fmtPair).join('\n')) | |
| console.log("\nworst matches:\n" + matches.reverse().slice(0, 20).map(fmtPair).join('\n')) | |
| } | |
| //zeta(0.5, 2002.896408374) | |
| //zeta(0.5, 1001.349482638); | |
| run(0, 2000); |
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Small experiment to see if the zeta function converges conditionally. It seems to be the case because the list of zeros that I am getting are a close match:
Program output:
Actual zeros (from http://www.dtc.umn.edu/~Odlyzko/zeta_tables/zeros1)
Of course, this program is very naive. It uses floating-point and standard math library. It would need to be rewritten with high precision libraries. But it already gives good results when the imaginary part is > 1000. With small values, it fails to find zeros but I suspect that this is because of a lack of precision (cancellation and thus convergence works better with high values because the terms rotate faster).
See http://math.stackexchange.com/questions/250470/conditional-convergence-of-riemanns-zetas-series/1774625#1774625