2c:null 2d:["$","title",null,{"children":"Hurwitz Zeta Partial Sums | ƒ | Observable"}] 32:T1818,{ const ζ_θ_trace = rotoshifted_detailed_traces.filter((t) => t.name === 'ζ_θ(s,t)')[0] const ζ_τ_trace = rotoshifted_detailed_traces.filter((t) => t.name === 'ζ^τ*(s,t)')[0] const ζ_θ_series = ζ_θ_trace.z.map((z,n) => ({ v: ℂ(ζ_θ_trace.x[n], ζ_θ_trace.y[n]), t: z+1 // add back the subtracted z offset })) const ζ_τ_series = ζ_τ_trace.z.map((z,n) => ({ v: ℂ(ζ_τ_trace.x[n], ζ_τ_trace.y[n]), t: z+1 // add back the subtracted z offset })) // NB this nearly tracks abs(ζ(0+yi,a)), but flipped... // const max = Math.max(...ζ_θ_trace.z) // const d_ds_series = [] // for (let t = 0; t <= max; t+= 0.1) { // d_ds_series.push({ // t, // v: 1/(π*t) * mathjs.log(mathjs.gamma(t-s.im/π)) - mathjs.log(2*π) / 2 // }) // } const max = Math.max(...ζ_θ_trace.z) const d_ds_series = [] for (let t = 0; t <= max; t+= 0.1) { // y: ζ_θ_series.map((o) => mathjs.eval('log(gamma(t)) - 1/2 log(2 pi)', o)), d_ds_series.push({ t, v: mathjs.log(mathjs.gamma(t-Math.abs(s.im)/π)) - mathjs.log(2*π) / 2 }) } const kink_t_low = Math.abs(s.im) / (π * hurwitz_spin_factor) const kink_t_low_arg = ζ_H(s,kink_t_low).mul(mathjs.exp(ℂ('i').mul(Math.abs(s.im)))).arg() // NB not right const kink_t_approx = Math.abs(s.im) / (π * hurwitz_spin_factor) + 1/2 // 1 - γ0 const kink_t_approx_arg = ζ_H(s,kink_t_approx).mul(mathjs.exp(ℂ('i').mul(π * hurwitz_spin_factor * kink_t_approx))).arg() const traces = [ { name: 'ℜ{ζ_θ(s,t)}', x: ζ_θ_series.map((o) => o.t), y: ζ_θ_series.map((o) => o.v.re), type: 'scatter', mode: 'lines', marker: { color: 'blue', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, { name: 'ℑ{ζ_θ(s,t)}', x: ζ_θ_series.map((o) => o.t), y: ζ_θ_series.map((o) => o.v.im), type: 'scatter', mode: 'lines', marker: { color: 'red', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, // abs/arg { name: '|ζ^θ(s,t)|', x: ζ_θ_series.map((o) => o.t), y: ζ_θ_series.map((o) => o.v.abs()), type: 'scatter', mode: 'lines', marker: { color: 'pink', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, { name: '-|ζ^θ(s,t)|', x: ζ_θ_series.map((o) => o.t), y: ζ_θ_series.map((o) => -o.v.abs()), type: 'scatter', mode: 'lines', marker: { color: 'pink', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, { name: 'Arg(ζ^θ(s,t))', x: ζ_θ_series.map((o) => o.t), y: ζ_θ_series.map((o) => o.v.arg()), type: 'scatter', mode: 'lines', marker: { color: 'purple', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, // { // name: '∂/∂s ζ(0,t), s=0', // x: d_ds_series.map((o) => o.t), // y: d_ds_series.map((o) => o.v), // type: 'scatter', // mode: 'lines', // marker: { // color: 'teal', // }, // visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', // }, // approximate maximal kink { name: `ζ_θ(s,ℑs/(πr)) inflection`, x: [ kink_t_low, kink_t_approx ], y: [ kink_t_low_arg, kink_t_approx_arg ], type: 'scatter', mode: 'markers', marker: { // size: 3, color: 'gray', }, visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, { name: 'ℜ{ζ^τ*(s,t)}', x: ζ_τ_series.map((o) => o.t), y: ζ_τ_series.map((o) => o.v.re), // .map((x,n) => ℂ(x,ζ_τ_trace.y[n]).div(χs).re), type: 'scatter', mode: 'lines', marker: { color: 'green', }, visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }, { name: 'ℑ{ζ^τ*(s,t)}', x: ζ_τ_series.map((o) => o.t), y: ζ_τ_series.map((o) => o.v.im), type: 'scatter', mode: 'lines', marker: { color: 'orange', }, visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }, { name: '|ζ^τ*(s,t)|', x: ζ_τ_series.map((o) => o.t), y: ζ_τ_series.map((o) => o.v.abs()), type: 'scatter', mode: 'lines', marker: { color: 'brown', }, visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }, { name: 'Arg(ζ^τ*(s,t))', x: ζ_τ_series.map((o) => o.t), y: ζ_τ_series.map((o) => o.v.arg()), type: 'scatter', mode: 'lines', marker: { color: 'tan', }, visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }, { name: `q point, |χs|`, x: [ q, q, q ], y: [ ζs_over_ζs_reflected.abs(), 0, -ζs_over_ζs_reflected.abs() ], type: 'scatter', mode: 'markers+lines', marker: { // size: 3, color: 'black', }, line: { dash: 'dot', }, // visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', }, ] const layout = { autosize: false, width: plot_width, height: plot_height, showlegend: true, xaxis: { // type: 'log', // rangemode: 'tozero', // range: [ 0, 95 ], }, yaxis: { autorange: false, range: [ -rval*2, rval*2 ], } } Plotly.react( graph2Div, traces, layout, { // scrollZoom: true, displayModeBar: true, displaylogo: false, // modeBarButtonsToRemove: ['toImage'], // responsive: true, }, ) }33:T55b,// neutered? viewof hurwitz_spin_factor = slider({ min: -4, max: 4, step: "any", value: 2, description: `hurwitz spin factor` }) // viewof hurwitz_spin_factor = html`` // hurwitz_spin_factor = (π+1)**(1/2) // setting to π-1 appears to force pendant center to approximate a log spiral? all the way down to s.im/2π // or at least something clearly not a euler spiral... wtf? // hurwitz_spin_factor = π-1 // √(π(π-1)) similarly // √(π+1) looks to do something even goofier... but could just be because it's very nearly 2 // hurwitz_spin_factor = Math.sqrt(π+1) // like a epicycloidal log spiral opening up... has neatly regular Arg function, like a fourier approximation to sawtooth function, mod(2πx,2π)-π... Re an approximation to -cos(2πx), Im like an approximation to -sin(2πx)... // though if you look closer at the initial range, wild a bit wilder, it is still contained... // the number of cycles before abs value shaft comes to a head related to height... // last full cycle appears to be ~ s.im/(π * 9/4) // Abs unchanged by e^iπ spin factor (as always)... as height s.im grows, area of approximamtion grows with it... // the surface upon which these approximations are drawn is...amusingly phallic // the girth of this phallis being χ(s) proves RH34:T585,ζs_rotoshift = (t, z_t) => { const ζs_t_origin = z_t || ζ_H(s, t) const scalefrac = t - Math.floor(t) // rotate all sums by e^(2π i frac(t)) // = e^(2π i (t - floor(t))) // = e^(2π i t - 2π i floor(t))) // = e^(2π i t) / e^(2π i floor(t)) const rotation = mathjs.exp(ℂ('i').mul(π * scalefrac * partials_folding_ratio)) let d = ℂ(t).pow(s_neg) d = ζs_t_origin.sub(d) // .mul(rotation) let xn = Math.abs(s.im) / (2*π*t) - 1/2 if (xn >= 0) d = d.mul(rotation) // this rotation is only factor for approach to C_1 pendant center // (ζ(s,t) - t^(-s)) e^(2π i t) / e^(2π i floor(t)) // t^(-s) = e^(t^(-s)/ln(t^-s)) = e^(t^(-s)/(-s ln(t)) ... // TODO show equality between χ-normalized hurwitz for s and 1-s through reflection // demonstrate that only way they could be equal is if |γr(s)| = |γr(1-s)| = 1 // this is known to be true iff Re(s) = 1/2 let k = 0 while (++k < xn) { // unshift by kth normalized reflected partial step, rotate, shift back // shift = ζn(1-s) χ(s) // = (ζ(1-s,1) - ζ(1-s,k)) ζ(s,1) / ζ(1-s,1) // = ζ(1-s,1) ζ(s,1) / ζ(1-s,1) - ζ(1-s,k) ζ(s,1) / ζ(1-s,1) // = (1 - ζ(1-s,k) ζ(s,1) / ζ(1-s,1) // = (1 - ζ(1-s,k) γr(s) // = (1 - ζ(1-s,k)) π^(s-1/2) Γ((1-s)/2) / Γ(s/2) const shift = ζn(s_neg.add(1), k).mul(χs) d = d.sub(shift).mul(rotation).add(shift) } return d }35:T5da,harmonic_hurwitz_spiral_traces = { // const approx_ // const spiral = (t, z) => mathjs.eval('exp(z*t/(π-2))', { t, z, π: Math.PI }) const spiral = (z,t) => mathjs.eval('exp(2*π*t*z)*χs', { t, z: ℂ(z), π: Math.PI, χs }) return [ { name: 'r ≈ ζ(s,1-Hn)', type: 'scatter3d', mode: 'lines+markers', marker: { color: 'blue', size: 2, opacity: 0.2, }, visible: trace_visibility.includes('harmonic_spirals') ? true : 'legendonly', ...(() => { const data = {} const xs = [] const ys = [] const zs = [] const abs = Math.hypot(1 - s.re, s.im) const arg = Math.atan2(s.im, 1 - s.re) + Math.PI const offset = ℂ(0) // ζs // for (let n = 0; n <= N_subdiv; ++n) { for (let n = -100; n <= 100; ++n) { const t = n/100 // 1/n // const A = (t)**(ζs.re) * abs // const B = -ζs.im * Math.log(t) + arg const A = t**(1 - s.re) / abs const B = -s.im * Math.log(t) + arg // data.x.push(offset.re + A * Math.cos(B)) // data.y.push(offset.im + A * Math.sin(B)) // const d = offset.add(spiral(s, t).mul(ζs_half)) // const d = offset.add(spiral(ℂ(1,1), t).mul(ζs_half)) const d = offset.add(spiral(s, t)) xs.push(d.re) ys.push(d.im) zs.push(0) } return { x: xs, y: ys, z: zs } })(), } ] }36:T1ec0,rotoshifted_detailed_traces = [ (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; const d = ζs_rotoshift(t, ℂ(v.x, v.y)); data.x.push(d.re); data.y.push(d.im); data.z.push(t); }); return { name: `ζ^τ(s,t) rotoshift`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 2, color: "violet", opacity: 0.5 }, line: { // dash: 'dot', }, visible: "legendonly" // visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }; })(), (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; const d = ζs_rotoshift(t, ℂ(v.x, v.y)); // data.x.push(d.re) // data.y.push(d.im) // NB flip it back to align w/ original zeta partials, just because data.x.push(ζs.re - d.re); data.y.push(ζs.im - d.im); data.z.push(t); }); return { name: `ζ(s) - ζ^τ(s,t)`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 2, color: "maroon", opacity: 0.5 }, visible: trace_visibility.includes("rotoshifted_spirals_reversed") ? true : "legendonly" }; })(), (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; let d = ζs_rotoshift_smooth(t, ℂ(v.x, v.y)); if (trace_config.includes("normalize_rotated_spirals")) d = d.mul(γr_r).mul(mathjs.exp(ℂ("i").mul(π / 4))); data.x.push(d.re); data.y.push(d.im); data.z.push(t); }); return { name: `ζ^τ*(s,t)`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 2, color: "green", opacity: 0.5 }, visible: trace_visibility.includes("rotoshifted_spirals") ? true : "legendonly" }; })(), (() => { const v = ζs_rotoshift_smooth(q_kink); const data = { x: [v.re], y: [v.im], z: [q_kink] }; return { name: `ζ^τ*(s,q_kink)`, ...data, type: "scatter3d", mode: "markers", marker: { size: 4, color: "orange", opacity: 0.9 }, visible: trace_visibility.includes("rotoshifted_spirals") ? true : "legendonly" }; })(), { name: `C₀ spread`, ...(() => { const x = []; const y = []; const z = []; const text = []; const t = Math.abs(s.im) / π; const scalefrac = t - Math.floor(t); const rotation = mathjs.exp(ℂ("i").mul(π * scalefrac)); const C0 = C0_upper; const C0_ = C0_lower; // .mul(mathjs.exp(ℂ('i').mul(-C0_lower.arg()))) const diff = C0_.neg().arg() - C0.arg(); x.push(C0.re); y.push(C0.im); z.push(Math.abs(s.im) / π); text.push(C0.arg()); x.push(C0_.re); y.push(C0_.im); z.push(Math.abs(s.im) / π); text.push(C0_.arg() + " diff: " + diff); return { x, y, z, text }; })(), type: "scatter3d", mode: "markers", marker: { size: 3, color: "black", opacity: 0.7 }, visible: "legendonly" // trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', }, (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; let d = ℂ(v.x, v.y).mul( mathjs.exp(ℂ("i").mul(π * hurwitz_spin_factor * t)) ); if (trace_config.includes("normalize_rotated_spirals")) d = d.mul(γr_r).mul(mathjs.exp(ℂ("i").mul(π / 4))); data.x.push(d.re); data.y.push(d.im); data.z.push(t - 1); }); return { name: `ζ_θ(s,t)`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 1, color: "gray" // opacity: 0.5, }, // visible: 'legendonly', visible: trace_visibility.includes("rotated_hurwitz_spirals") ? true : "legendonly" }; })(), (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; const d = ℂ(v.x, v.y).mul(mathjs.exp(ℂ("i").mul(π * t))); data.x.push(d.re); data.y.push(d.im); data.z.push(t - 1); }); return { name: `ζ_θ[π](s,t)`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 1, color: "rgba(0,0,200,0.2)" // opacity: 0.5, }, // visible: 'legendonly', visible: trace_visibility.includes("fixed_half_hurwitz_spirals") ? true : "legendonly" }; })(), (() => { const data = { x: [], y: [], z: [] }; ζs_H_trace.forEach((v) => { const t = v.z; if (t < 1) return; if (rotoshift_partial_min < Math.abs(s.im) / (t * π)) return; const d = ℂ(v.x, v.y).mul(mathjs.exp(ℂ("i").mul(π * t + π))); data.x.push(d.re); data.y.push(d.im); data.z.push(t - 1); }); return { name: `ζ_θ[π*](s,t)`, ...data, type: "scatter3d", mode: "lines+markers", marker: { size: 1, color: "rgba(200,0,0,0.2)" // opacity: 0.5, }, // visible: 'legendonly', visible: trace_visibility.includes("fixed_half_hurwitz_spirals") ? true : "legendonly" }; })(), (() => { const t = Math.abs(s.im) / (π * hurwitz_spin_factor); const p = ζ_H(s, t).mul( mathjs.exp(ℂ("i").mul(π * hurwitz_spin_factor * t)) ); const data = { x: [0, p.re], y: [0, p.im], z: [t - 1, t - 1], text: ["ζ(s,θ)"] }; return { name: `ζ_θ(s,ℑs/(πr)) inflection`, ...data, type: "scatter3d", mode: "markers+lines+text", marker: { size: 3, color: "gray" }, line: { dash: "dot" }, visible: trace_visibility.includes("rotated_hurwitz_spirals") ? true : "legendonly" }; })() // (() => { // const range = [10,45] // const data = { x: [], y: [], z: [] } // linspace(range[0], range[1], 201).forEach(t => { // const d = ζs_rotoshift(t) // data.x.push(d.re) // data.y.push(d.im) // data.z.push(t) // }) // return { // name: `ζs_q:[${range}]_rotoshift`, // ...data, // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 3, // color: 'violet', // }, // line: { // dash: 'dot', // } // } // })(), // (() => { // const range = [1,10] // const data = { x: [], y: [], z: [] } // linspace(range[0], range[1], 201).forEach(t => { // const d = ζs_rotoshift(t) // data.x.push(d.re) // data.y.push(d.im) // data.z.push(t) // }) // return { // name: `ζs_q:[${range}]_rotoshift`, // ...data, // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 3, // color: 'red', // }, // line: { // dash: 'dot', // } // } // })() ]37:T1915,rotoshifted_hurwitz_traces = { const T = rotoshift_num / rotoshift_denom const ζs_a_origin = ζ_H(s, T) const scalefrac = T - Math.floor(T) // rotate all sums by e^(2π * frac(T)) const rotation = mathjs.exp(ℂ('i').mul(Math.PI).mul(scalefrac)) return [ buildCurve( 'ζn^τ(s,a): C_n-rotoshifted/rotated', { marker: { color: 'blue', size: 2, }, line: 'blue', visible: 'legendonly', }, (() => { const n = 1 // pendant target const p_index_range = [ Math.abs(s.im) / (2 * n * Math.PI + Math.PI), Math.abs(s.im) / (2 * n * Math.PI - Math.PI) ] const discrete_range = [ Math.ceil(p_index_range[0]), Math.floor(p_index_range[1]) ] // calculate hurwitz partials... const L = [ℂ(0)] const vals = {x:[],y:[],z:[]} for (let i = 1; i <= discrete_range[1]; ++i) { const t = i - 1 + T let d = L[i] = L[i-1].add(ℂ(t).pow(s_neg)) d = ζs_a_origin.sub(d).mul(rotation) let xn = Math.abs(s.im) / (2 * Math.PI * (t + 1/2) - Math.PI) - 1/2 // if (xn < 0) ... TODO turn around... appears we're not taking shortest path here let k = 0 while (++k < xn) { // shift one normalized/reversed partial step, rotate, shift back const shift = ζn(s_neg.add(1), k).mul(ζs).div(ζr) d = d.sub(shift).mul(rotation).add(shift) } vals.x.push(d.re) vals.y.push(d.im) vals.z.push(t) } return vals })() ), buildCurve( 'ζn(s,a): C₁-rotoshifted,rotated', { marker: { color: 'brown', size: 2, }, line: 'brown', visible: 'legendonly', }, (() => { const n = 1 // pendant target const p_index_range = [ Math.abs(s.im) / (2 * n * Math.PI + Math.PI), Math.abs(s.im) / (2 * n * Math.PI - Math.PI) ] const discrete_range = [ Math.ceil(p_index_range[0]), Math.floor(p_index_range[1]) ] // calculate hurwitz partials... const L = [ℂ(0)] const vals = {x:[],y:[],z:[]} for (let i = 1; i <= discrete_range[1]; ++i) { const t = i-1 let d = L[i] = L[i-1].add(ℂ(t + T).pow(s_neg)) // rescale by rotation factor d = ζs_a_origin.sub(d).mul(rotation) vals.x.push(d.re) vals.y.push(d.im) vals.z.push(t + T) } return vals })() ), buildCurve( 'ζn(s,a): C₁-rotoshifted', { // marker: { // color: 'purple', // size: 2, // }, line: 'purple', visible: 'legendonly', }, (() => { // partial sum halfway between pendant center C_n and C_[n-1]: // Math.abs(s.im) / (2 * n * Math.PI) // partial sum approaching pendant center C_n from below: // Math.abs(s.im) / (2 * n * Math.PI + Math.PI) const n = 1 // pendant target // range gives indices for partial sums falling into nth specific pendant // TODO abs value on Im(s)? const p_index_range = [ Math.abs(s.im) / (2 * n * Math.PI + Math.PI), Math.abs(s.im) / (2 * n * Math.PI - Math.PI) ] // zeroth pendant is log spiral approached by hurwitz zeta aboout zero // first pendant is final curl approaching zeta -- magnitude: ζ(s)/ζ(1-s) // second pendant magnitude: 2^s ζ(s)/ζ(1-s) ??? const discrete_range = [ Math.ceil(p_index_range[0]), Math.floor(p_index_range[1]) ] // calculate hurwitz partials... const L = [ℂ(0)] const vals = {x:[],y:[],z:[]} for (let i = 1; i <= discrete_range[1]; ++i) { const t = i-1 let d = L[i] = L[i-1].add(ℂ(t + T).pow(s_neg)) d = ζs_a_origin.sub(d) vals.x.push(d.re) vals.y.push(d.im) vals.z.push(t + T) } return vals })() ), // { // name: `ζ^τ(s,q+1) point, ∂/∂α vector`, // x: [ ζs_q_rotoshift.re, ζs_q_rotoshift.re + ζs_q_rotoshift_dq.re ], // y: [ ζs_q_rotoshift.im, ζs_q_rotoshift.im + ζs_q_rotoshift_dq.im ], // z: [ q, q ], // type: 'scatter3d', // mode: 'markers+lines', // visible: trace_visibility.includes('rotoshifted_spirals') ? true : 'legendonly', // marker: { // size: 3, // color: 'blue', // opacity: 1, // }, // line: { // color: 'black', // } // }, (() => { const r = hurwitz_spin_factor const ζ_base = ζ_H(s,q+1) const p = ζ_base.mul(mathjs.exp(ℂ('i').mul(π * r * q))) // d/da (ζ(s,a) exp(iπra)) = (-iπr ζ(s,a) + ζ(s+1,a) * s) * exp(iπra) const ζs_q_spin_dq = mathjs.exp(ℂ('i').mul(π*q*r)).mul( ℂ('i').mul(π*r).mul(ζs_q_next).add(ζs_q_next_dq) ) const dq = ζs_q_spin_dq const dv = p.add(ζs_q_spin_dq) // normalize dq for tan vector const Tv = ζs_q_spin_dq.div(ζs_q_spin_dq.abs()) const Tp = p.add(Tv) const data = { x: [ dv.re, p.re, Tp.re ], y: [ dv.im, p.im, Tp.im ], z: [ q + 1, q, q ], text: [ '', `ζ_θ(s,${q+1}) tan` ], } return { name: `ζ_θ(s,q+1) tangent vector`, ...data, type: 'scatter3d', mode: 'markers+lines', visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', marker: { size: 3, color: 'red', opacity: 1, }, line: { color: 'red', } } })(), // (() => { // const p = mathjs.eval('q^(1-s) / (s-1) + 1/2 q^(-s)', {s,q}) // const data = { // x: [ 0, p.re ], // y: [ 0, p.im ], // z: [ q, q ], // } // return { // name: `hurwitz integral offset`, // ...data, // type: 'scatter3d', // mode: 'markers+lines', // visible: trace_visibility.includes('rotated_hurwitz_spirals') ? true : 'legendonly', // marker: { // size: 3, // color: 'green', // opacity: 1, // }, // line: { // color: 'green', // } // } // })(), ] }38:T5be,/* (d^1)/(dq^1)(ζ(s,q) exp(i π q r)) = exp(i π q r) * ( + 1 (-i π r)^1 ζ(s,q) + 1 (-i π r)^0 ζ(s+1,q) (s) ) (d^2)/(dq^2)(ζ(s,q) exp(i π q r)) = exp(i π q r) * ( + 1 (-i π r)^2 ζ(s+0,q) + 2 (-i π r)^1 ζ(s+1,q) (s) + 1 (-i π r)^0 ζ(s+2,q) (s)(s+1) ) (d^3)/(dq^3)(ζ(s,q) exp(i π q r)) = exp(i π q r) * ( + 1 (-i π r)^3 ζ(s+0,q) + 3 (-i π r)^2 ζ(s+1,q) (s) + 3 (-i π r)^1 ζ(s+2,q) (s)(s+1) + 1 (-i π r)^0 ζ(s+3,q) (s)(s+1)(s+2) ) (d^4)/(dq^4)(ζ(s,q) exp(i π q r)) = exp(i π q r) * ( + 1 (-i π r)^4 ζ(s+0,q) + 4 (-i π r)^3 ζ(s+1,q) (s) + 6 (-i π r)^2 ζ(s+2,q) (s)(s+1) + 4 (-i π r)^1 ζ(s+3,q) (s)(s+1)(s+2) + 1 (-i π r)^0 ζ(s+4,q) (s)(s+1)(s+2)(s+3) ) (d^k)/(dq^k)(ζ(s,q) exp(i π q r)) = exp(i π q r) ∑[n=0 to k] binom(n+1,k+1) (-i π r)^n ζ(s+n,q) s^((n)) where s^((n)) notation represents rising factorial s^((n)) = Γ(s+n)/Γ(s) binom(n,k) = n! / (k! * (n-k)!) for non-negative integers... for any n: binomial(-n,k) = (-1)^k binomial(n+k-1,k) binomial(n,k) = (-1)^k binomial(k-n-1,k) = (-1)^k (k-n-1)! / (k! (n-1)!) = exp(i π q r) ∑[n=0 to k] binom(n+1,k+1) (-i π r)^n ζ(s+n,q) Γ(s+n)/Γ(s) integral: (d^-1)/(dq^-1) (ζ(s,q) exp(i π q r)) = exp(i π q r) ( + binom(0,1) (-i π r)^-1 ζ(s+0,q) + binom(0,0) (-i π r)^-1 ζ(s-1,q) s^((-1)) ) = exp(i π q r) ζ(s-1,q) s^((-1)) / (-i π r) */39:T72b,pendant_spool_traces = { const length = Math.min(10, Math.floor(Math.abs(s.im) / (3 * Math.PI))) // iterate over reflected partials, rescale by ζ(1-s)/ζ(s) const rpartials = Array.from({ length }).map((_,i) => ( ζn(s_neg.add(1), i).mul(ζs).div(ζr) )) // const rhpartials = Array.from({ length: 3 }).map((_,i) => { // const n = Math.floor((i+1)/2) // const h = Math.abs(s.im) / (2*n * Math.PI) // return { h, v: ζs_rotoshift(h) } // }) const C_name = (n) => `C_${n}` return rpartials.map((z,i) => ({ name: C_name(i), x: [ z.re, z.re, z.re ], y: [ z.im, z.im, z.im ], z: [ Math.abs(s.im) / ((2*i+2)*Math.PI), Math.abs(s.im) / ((2*i+1)*Math.PI), i == 0 ? (N+1) : (Math.abs(s.im) / ((2*i)*Math.PI)) ], text: [ '', C_name(i), '' ], type: 'scatter3d', mode: 'markers+lines+text', marker: { size: 2, color: 'black', // opacity: 0.6, } })) .concat([ { name: `C_n`, x: rpartials.map(v => [ v.re, v.re ]).flat(1), y: rpartials.map(v => [ v.im, v.im ]).flat(1), z: rpartials.map((_,i) => { const n = Math.floor((i+1)/2) return Math.abs(s.im) / (2*n * Math.PI) }), type: 'scatter3d', mode: 'markers+lines', marker: { size: 2, color: 'black', // opacity: 0.6, }, line: { dash: 'dot', }, }, // { // name: `H_n midpoints`, // x: rhpartials.map(({v}) => v.re), // y: rhpartials.map(({v}) => v.im), // z: rpartials.map(({h}) => h), // type: 'scatter3d', // mode: 'markers+lines', // marker: { // size: 2, // color: 'green', // // opacity: 0.6, // }, // line: { // // dash: 'dot', // }, // }, ]) }3a:T7b0,spiral_approximation_traces = [ buildCurve( 'r(θ) ≈ ζ(s)', { line: { color: 'purple', // color: 'rgba(80,0,20,0.75)', // shape: 'spline', // smoothing: 1.3, // dash: 'dot', }, visible: trace_visibility.includes('approx_spirals') ? true : 'legendonly', }, (() => { const length = 1 + N * (1 + N_subdiv) const data = {} data.x = Array(length) data.y = Array(length) data.z = Array(length) const abs = Math.hypot(1 - s.re, s.im) const arg = Math.atan2(s.im, 1 - s.re) const offset = ζs const space = linspace(0, 1 + N, 1 + 5 * (1 + N)) space.shift() // drop first entry space.forEach((t,i) => { const A = t**(1 - s.re) / abs const B = -s.im * Math.log(t) + arg data.x[i] = offset.re + A * Math.cos(B) data.y[i] = offset.im + A * Math.sin(B) data.z[i] = t - 1/2 }) return data })() ), buildCurve( 'r(θ) - ζ(s) ≈ 0', { line: { color: 'green', // color: 'rgba(80,0,20,0.75)', // shape: 'spline', // smoothing: 1.3, // dash: 'dot', }, visible: trace_visibility.includes('approx_spirals') ? true : 'legendonly', }, (() => { const length = 1 + N * (1 + N_subdiv) const data = {} data.x = Array(length) data.y = Array(length) data.z = Array(length) const abs = Math.hypot(1 - s.re, s.im) const arg = Math.atan2(s.im, 1 - s.re) const offset = ℂ(0) // ζs const space = linspace(0, 1 + N, 1 + 5 * (1 + N)) space.shift() // drop first entry space.forEach((t,i) => { const A = -1 * t**(1 - s.re) / abs const B = -s.im * Math.log(t) + arg data.x[i] = offset.re + A * Math.cos(B) data.y[i] = offset.im + A * Math.sin(B) data.z[i] = t - 1/2 }) return data })() ) ]3b:T4d8,symmetry_spiral_traces = [ buildCurve( 'ζn(s)/ζn(1-s) ≤ t ≤ ζn(1-s)/ζn(s)', { line: { color: 'rgba(20,20,20,0.5)', shape: 'spline', smoothing: 1, // dash: 'dot', }, visible: 'legendonly', }, (() => { // const length = 1 + N * (1 + N_subdiv) const length = 75 const data = {} data.x = Array(length) data.y = Array(length) data.z = Array.from({ length }).fill(0) data.text = Array(length) // (|Z(1-s)|/|ζ(s)|)^-t cis(-t(Arg(Z(1-s)) - Arg(Z(s))) // (|Z(s)|/|ζ(1-s)|)^t cis(t(Arg(Z(s)) - Arg(Z(1-s))) const ζabs_ratio = ζs.abs() / ζr.abs() const ζarg_ratio = ζs.arg() - ζr.arg() linspace(-1, 1, length).forEach((t,i) => { const ζabs_ratio_t = ζabs_ratio ** t const ζarg_ratio_t = ζarg_ratio * t data.x[i] = ζabs_ratio_t * Math.cos(ζarg_ratio_t) data.y[i] = ζabs_ratio_t * Math.sin(ζarg_ratio_t) if (i === 0) { data.text[i] = 'ζn(1-s)ζn(s))' } else if (i === length - 1) { data.text[i] = 'ζn(s)/ζn(1-s)' } else { // data.text[i] = '' } }) return data })() ) ]3c:T62c,hurwitz_traces = [ { name: `ζ(s,t)`, x: ζs_H_trace.map(v => v.x), y: ζs_H_trace.map(v => v.y), z: ζs_H_trace.map(v => v.z - 1), type: 'scatter3d', mode: 'lines', hoverinfo: 'none', visible: trace_visibility.includes('hurwitz_spirals') ? true : 'legendonly', // marker: { // color: 'orange', // size: 2, // opacity: 0.6, // }, line: { simplify: false, color: 'rgba(255,105,0,0.6)', }, }, { name: `ζ(s,q+1) point, ∂/∂α vector`, x: [ ζs_q_next.re, ζs_q_next.re + ζs_q_next_dq.re ], y: [ ζs_q_next.im, ζs_q_next.im + ζs_q_next_dq.im ], z: [ q, q ], text: [ 'arg(∂/∂α ζ(s,q+1)): ' + ζs_q_next_dq.arg(), '|∂/∂α ζ(s,q+1)|: ' + ζs_q_next_dq.abs() ], type: 'scatter3d', mode: 'markers+lines', visible: trace_visibility.includes('hurwitz_spirals') ? true : 'legendonly', marker: { size: 3, color: 'red', opacity: 0.8, }, }, { name: `ζ(s,t) : t ≤ 1`, x: ζs_H_trace.map(v => v.z <= 1 ? v.x : null).filter(k => k != null), y: ζs_H_trace.map(v => v.z <= 1 ? v.y : null).filter(k => k != null), z: ζs_H_trace.map(v => v.z <= 1 ? v.z - 1 : null).filter(k => k != null), type: 'scatter3d', mode: 'lines', // marker: { // color: 'orange', // size: 2, // opacity: 0.6, // }, // hoverinfo: 'none', visible: trace_visibility.includes('harmonic_spirals') ? true : 'legendonly', line: { simplify: false, color: 'rgba(255,105,0,0.9)', }, }, ]3d:T2484,traces = [ ...axis_traces, ...hurwitz_traces, ...symmetry_spiral_traces, ...spiral_approximation_traces, ...rotoshifted_hurwitz_traces, ...rotoshifted_detailed_traces, ...pendant_spool_traces, ...harmonic_hurwitz_spiral_traces, // { // // TODO this isn't right -- we actually need to shift to origin, then apply transform, then unshift} // name: `rotoscaled ζn(s)`, // x: ζs_partials.map(v => v.pow(reflected_rotoscale).re), // y: ζs_partials.map(v => v.pow(reflected_rotoscale).im), // z: ζs_partials.map((v,n) => n), // mode: 'lines+markers', // type: 'scatter3d', // marker: { // color: 'red', // size: 2, // opacity: 0.6, // }, // }, { name: `ζn(s)`, x: ζs_partials.map(v => v.re), y: ζs_partials.map(v => v.im), z: ζs_partials.map((v,n) => n), mode: 'lines+markers', type: 'scatter3d', // hoverinfo: 'x+y', marker: { color: 'red', size: 2, opacity: 0.6, }, line: { color: 'rgba(150,0,0,0.6)', }, }, { name: `ζn(1-s)`, x: ζr_partials.map(v => v.re), y: ζr_partials.map(v => v.im), z: ζr_partials.map((v,n) => n), mode: 'lines+markers', type: 'scatter3d', hoverinfo: 'x+y', marker: { color: 'pink', size: 2, opacity: 0.6, }, visible: 'legendonly', }, { name: `ζn(1-s) : 1/z`, x: ζr_partials.map(v => v.re), y: ζr_partials.map(v => v.im), z: ζr_partials.map((_,n) => 1/n), mode: 'lines+markers', type: 'scatter3d', marker: { color: 'pink', size: 2, opacity: 0.6, }, // visible: 'legendonly', }, { name: `ζ(s) - ζn(s)`, x: ζs_partials.map(v => ζs.re - v.re), y: ζs_partials.map(v => ζs.im - v.im), z: ζs_partials.map((v,n) => n), mode: 'lines+markers', type: 'scatter3d', marker: { color: 'steelblue', size: 2, opacity: 0.6, }, line: { color: 'rgba(40,100,140,0.6)', // dash: 'dot', }, }, { name: `ζn(1-s) γr_s, rescaled`, x: ζr_partials.map(v => v.mul(γr_s).re), y: ζr_partials.map(v => v.mul(γr_s).im), // maps n=0 to s.im/π, n=1 to s.im/(3π), etc... z: ζr_partials.map((v,n) => Math.abs(s.im) / ((2*n+1)*π)), mode: 'lines+markers', type: 'scatter3d', marker: { color: 'magenta', size: 2, opacity: 0.6, }, visible: trace_visibility.includes('rescaled_zeta_partials') ? true : 'legendonly', }, // { // name: `ζs-ζ(s,qt)`, // x: ζ_H_trace[0].map((v) => ζs[0] - v), // y: ζ_H_trace[1].map((v) => ζs[1] - v), // z: ζ_H_trace[2].map((v) => v-1), // type: 'scatter3d', // mode: 'lines', // line: { // color: 'rgba(0,105,255,0.6)', // simplify: false, // }, // }, // { // name: `ζn_q(s,q)`, // // TODO reverse these partial sums such that they're centered on pole // x: ζs_q_partials.x, // y: ζs_q_partials.y, // z: ζs_q_partials.z.map(() => q - 1), // type: 'scatter3d', // mode: 'lines', // line: { // simplify: false, // color: 'rgba(0, 48, 0, 0.6)', // }, // }, // { // name: `ζn(s,q)`, // x: ζs_q_partials.x, // y: ζs_q_partials.y, // z: ζs_q_partials.z.map((v) => v + q - 1), // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 2, // opacity: 0.6, // }, // line: { // simplify: false, // color: ζs_q_partials.z, // colorscale: 'Greens', // }, // }, // { // name: `ζ(s)-ζ(s,q)+ζn(s,q)`, // x: ζs_q_partials.x.map((v) => (ζs[0] - ζs_q.re + v)), // y: ζs_q_partials.y.map((v) => (ζs[1] - ζs_q.im + v)), // z: ζs_q_partials.z.map((v) => v + q - 1), // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 2, // opacity: 0.6, // }, // line: { // color: ζs_q_partials.z, // colorscale: 'Viridis', // }, // }, // { // name: `ζ(s) - ζ(s, ℑs / π)`, // x: [ ζs[0] - ζs_ℑs_π[0] ], // y: [ ζs[1] - ζs_ℑs_π[1] ], // z: [ s[1] / Math.PI - 1 ], // text: [ 'ζ(s) - ζ(s, ℑs / π)' ], // type: 'scatter3d', // mode: 'markers+text', // marker: { // size: 3, // color: 'steelblue', // // opacity: 0.6, // }, // line: { // simplify: false, // color: 'rgba(70, 130, 180, 0.8)', // }, // }, // { // name: `ζ(s)-ζ(s,q)+ζn_q(s,q)`, // x: ζs_q_partials.x.map((v) => (ζs[0] - ζs_q.re + v)), // y: ζs_q_partials.y.map((v) => (ζs[1] - ζs_q.im + v)), // z: ζs_q_partials.z.map(() => q - 1), // type: 'scatter3d', // mode: 'lines', // line: { // simplify: false, // color: 'rgba(60, 12, 86, 0.6)', // }, // }, // { // name: `ζ(s, ℑs / π)`, // x: [ ζs_ℑs_π[0] ], // y: [ ζs_ℑs_π[1] ], // z: [ s[1] / Math.PI - 1 ], // text: [ 'ζ(s, ℑs / π)' ], // type: 'scatter3d', // mode: 'markers+text', // marker: { // size: 3, // color: 'orange', // // opacity: 0.6, // }, // line: { // simplify: false, // color: 'rgba(70, 130, 180, 0.8)', // }, // }, // { // name: `ζ(s,q)`, // x: [ ζs_q.re, ζs_q.re, ζs_q.re ], // y: [ ζs_q.im, ζs_q.im, ζs_q.im ], // z: [ 1, q-1, N+1 ], // text: ['','','ζ(s,q)'], // type: 'scatter3d', // mode: 'markers+lines+text', // marker: { // size: 3, // color: 'green', // opacity: 0.6, // }, // line: { // simplify: false, // color: 'rgba(0, 48, 0, 0.4)', // }, // }, // (() => { // const data = {} // data.x = ζs_q_partials.map((v) => (v.re)) // data.y = ζs_q_partials.map((v) => (v.im)) // data.z = ζs_q_partials.map(() => q) // return { // name: `ζn(s,q)`, // ...data, // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 1, // opacity: 0.5, // color: 'lime', // }, // } // })(), // (() => { // const spin = mathjs.exp(ℂ('i').mul(π * hurwitz_spin_factor * q)) // const ζs_qθ_partials = ζs_q_partials.map((v) => ζs_q.mul(spin).sub(v.mul(spin))) // const data = {} // data.x = ζs_qθ_partials.map((v) => v.re) // data.y = ζs_qθ_partials.map((v) => v.im) // data.z = ζs_qθ_partials.map(() => q) // return { // name: `ζ_θ(s,q) - ζn_θ(s,q)`, // ...data, // type: 'scatter3d', // mode: 'lines+markers', // marker: { // size: 1, // opacity: 0.5, // color: 'coral', // }, // } // })(), (() => { const ζs_qθ_partials = ζs_q_partials.map((v) => v.mul(mathjs.exp(ℂ('i').mul(π * hurwitz_spin_factor * q)))) const data = {} data.x = ζs_qθ_partials.map((v) => v.re) data.y = ζs_qθ_partials.map((v) => v.im) data.z = ζs_qθ_partials.map(() => q) return { name: `ζn_θ(s,q)`, ...data, type: 'scatter3d', mode: 'lines+markers', marker: { size: 1, opacity: 0.5, color: 'olive', }, } })(), { name: `ζ(s)`, x: [ ζs.re, ζs.re, ζs.re, ζs.re ], y: [ ζs.im, ζs.im, ζs.im, ζs.im ], z: [ 0, 1, Math.abs(s.im) / Math.PI, N+1 ], text: ['', '', /*'ℑs / π'*/, 'ζ(s)'], type: 'scatter3d', mode: 'lines+markers+text', marker: { size: 3, color: 'steelblue', opacity: 0.6, }, line: { simplify: false, color: 'rgba(200, 10, 10, 0.4)', }, }, { name: `ζ(s,1/2)`, x: [ ζs.mul(ℂ(2).pow(s).sub(1)).re ], y: [ ζs.mul(ℂ(2).pow(s).sub(1)).im ], z: [ -1/2 ], text: [ 'ζ(s,1/2)' ], type: 'scatter3d', mode: 'markers+text', marker: { size: 3, color: 'black', opacity: 0.6, }, }, { name: `ζ(s,1/N_div)`, // x: [ ζs.mul(ℂ(2).pow(s).sub(1)).mul(ℂ(2).pow(s)).re ], // y: [ ζs.mul(ℂ(2).pow(s).sub(1)).mul(ℂ(2).pow(s)).im ], ...(() => { const v = ζ_H(s, 1/(N_subdiv + 1)) return { x: [ v.re ], y: [ v.im ] } })(), z: [ 1/(N_subdiv + 1) - 1 ], text: [ `ζ(s,1/${N_subdiv + 1})` ], type: 'scatter3d', mode: 'markers+text', marker: { size: 3, color: 'black', opacity: 0.6, }, visible: 'legendonly', }, { name: `ζ(s,q) -> origin`, x: [ ζs_q.re, 0 ], y: [ ζs_q.im, 0 ], z: [ q - 1, q - 1 ], type: 'scatter3d', mode: 'lines', line: { color: 'blue', }, visible: 'legendonly', }, { name: `ζ(s,q±1)`, x: [ ζs_q_prev.re, ζs_q.re, ζs_q_next.re ], y: [ ζs_q_prev.im, ζs_q.im, ζs_q_next.im ], z: [ q - 2, q - 1, q ], text: [ '', `ζ(s,q)`, '' ], type: 'scatter3d', mode: 'lines+markers+text', // hoverinfo: 'x+y', marker: { size: 3, color: 'rgba(226,140,18,0.9)', }, visible: 'legendonly', }, { name: `ζ(1-s)`, x: [ ζr.re ], y: [ ζr.im ], z: [ -1 ], text: [ `ζ(1-s)` ], type: 'scatter3d', mode: 'markers+text', marker: { size: 3, color: 'pink', opacity: 0.6, }, }, ]3e:T471,layout = ({ autosize: false, width: plot_width, height: plot_height, showlegend: true, // legend: { // x: 1, // y: 0.5 // // orientation: 'h', // }, title: "ζ(s,q) partial sum geometry", scene: { dragmode: "turntable", hovermode: "closest", aspectratio: { x: width_aspect_scale, y: width_aspect_scale, z: height_aspect_scale }, // annotations: https://plot.ly/javascript/reference/#layout-scene-annotations xaxis: { range: [-rval, rval], fixedrange: true, hoverformat: ".5r" // d3-format spec // showticklabels: false, }, yaxis: { range: [-rval, rval], fixedrange: true, hoverformat: ".5r" // d3-format spec // showticklabels: false, }, zaxis: { range: [z_range_min, z_range_max], fixedrange: true, showticklabels: false, type: z_axis_log_mode ? "log" : "linear" // ticks: 'outside', // nticks: 0, // exponentformat: 'e', // rangemode: 'tozero', }, margin: { l: 0, r: 0, b: 0, t: 0 }, camera: camera_current } })3f:T16d0,// from http://www.dtc.umn.edu/~odlyZko/zeta_tables/zeros2 ρ_i = [ 14.13472514173469379045725198356247027078425711569924, 21.02203963877155499262847959389690277733434052490278, 25.01085758014568876321379099256282181865954967255799, 30.42487612585951321031189753058409132018156002371544, 32.93506158773918969066236896407490348881271560351703, 37.58617815882567125721776348070533282140559735083079, 40.91871901214749518739812691463325439572616596277727, 43.32707328091499951949612216540680578264566837183687, 48.00515088116715972794247274942751604168684400114442, 49.77383247767230218191678467856372405772317829967666, 52.97032147771446064414729660888099006382501788882122, 56.44624769706339480436775947670612755278226447171663, 59.34704400260235307965364867499221903109877280646666, 60.83177852460980984425990182452400380291009045121917, 65.11254404808160666087505425318370502934814929516672, 67.07981052949417371447882889652221677010714495174555, 69.54640171117397925292685752655473844301247420960251, 72.06715767448190758252210796982616839048090662145669, 75.70469069908393316832691676203034592281190353069740, 77.14484006887480537268266485630463701579603244923446, 79.33737502024936792276359287711622819061324674312003, 82.91038085408603018316483749477060949750888059378214, 84.73549298051705010573531120682774141710662793424081, 87.42527461312522940653166785091921325217188640126902, 88.80911120763446542368234807950937839544489340981867, 92.49189927055848429625972524181068487872179402773064, 94.65134404051988696659792581520815393772802701565485, 95.87063422824530975874102921924678169525646122498799, 98.83119421819369223332442013862232782065803906342819, 101.31785100573139122878544794029230890633286638430089, 103.72553804047833941639840810869528083448117306949576, 105.44662305232609449367083241411180899728275392853513, 107.16861118427640751512335196308619121347670788140476, 111.02953554316967452465645030994435041534596839007305, 111.87465917699263708561207871677059496031174987338587, 114.32022091545271276589093727619107980991765772382989, 116.22668032085755438216080431206475512732985123238322, 118.79078286597621732297913970269982434730621059280938, 121.37012500242064591894553297049992272300131063172874, 122.94682929355258820081746033077001649621438987386351, 124.25681855434576718473200796612992444157353877469356, 127.51668387959649512427932376690607626808830988155498, 129.57870419995605098576803390617997360864095326465943, 131.08768853093265672356637246150134905920354750297504, 133.49773720299758645013049204264060766497417494390467, 134.75650975337387133132606415716973617839606861364716, 138.11604205453344320019155519028244785983527462414623, 139.73620895212138895045004652338246084679005256538260, 141.12370740402112376194035381847535509030066087974762, 143.11184580762063273940512386891392996623310243035463, 146.00098248676551854740250759642468242897574123309580, 147.42276534255960204952118501043150616877277525047683, 150.05352042078488035143246723695937062303732155952820, 150.92525761224146676185252467830562760242677047299671, 153.02469381119889619825654425518544650859043490414550, 156.11290929423786756975018931016919474653530850094292, 157.59759181759405988753050315849876573072389951914173, 158.84998817142049872417499477554027141433508304942696, 161.18896413759602751943734412936955436491579032747546, 163.03070968718198724331103900068799489696446141647768, 165.53706918790041883003891935487479732836725174506860, 167.18443997817451344095775624621037873646076924261676, 169.09451541556882148950587118143183479666764858044162, 169.91197647941169896669984359582179228839443712534137, 173.41153651959155295984611864934559525415606606342011, 174.75419152336572581337876245586691793875571762057166, 176.44143429771041888889264105786093352811849710880971, 178.37740777609997728583093541418442618313236146127250, 179.91648402025699613934003661205123745368760755301840, 182.20707848436646191540703722698779869079745777823990, 184.87446784838750880096064661723425841335102291195066, 185.59878367770747146652770426839264661293471764951328, 187.22892258350185199164154058613124301681073460399031, 189.41615865601693708485228909984532449135710302319335, 192.02665636071378654728363142558343010583992029797709, 193.07972660384570404740220579437605460402061581054886, 195.26539667952923532146318781486225092690505245228692, 196.87648184095831694862226391469620773574602869194221, 198.01530967625191242491991870220886715506269543857099, 201.26475194370378873301613342754817322240286363918673, 202.49359451414053427768666063786431582102024489942005, 204.18967180310455433071643838631368513653452922874190, 205.39469720216328602521237939069309092372291477204840, 207.90625888780620986150196790775364426865940376888399, 209.57650971685625985283564428988675217539078318132616, 211.69086259536530756390748673071929425339403098293564, 213.34791935971266619063912202107260882189718327663306, 214.54704478349142322294420107259069104559988805308307, 216.16953850826370026586956335449812857545371427416411, 219.06759634902137898567725659043724124514918292701135, 220.71491883931400336911559263390633965676114507766196, 221.43070555469333873209747511927607795022233107731990, 224.00700025460433521172887552850489535608598994959552, 224.98332466958228750378252368052865677209005448558742, 227.42144427967929131046143616065963996396914832197662, 229.33741330552534810776008330605574008275234138781851, 231.25018870049916477380618677001037260670849584312337, 231.98723525318024860377166853919786220541983399456249, 233.69340417890830064070449473256978817953722775456583, 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Zeta Partial Sums","cells":[{"id":0,"value":"md`# Hurwitz Zeta Partial Sums`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1848,"value":"ζs_partials = {\n const L = [ℂ(0), ℂ(1)]\n for (let n = 2; n <= N; ++n) {\n L[n] = L[n-1].add(ℂ(n).pow(s_neg))\n }\n return L\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3315,"value":"ζr_partials = {\n const L = [ℂ(0), ℂ(1)]\n const z_neg = ℂ(1).sub(s).neg()\n for (let n = 2; n <= N; ++n) {\n L[n] = L[n-1].add(ℂ(n).pow(z_neg))\n }\n return L\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2175,"value":"frac_parts = {\n const denom = N_subdiv + 1\n const unit = 1 / denom\n const domain = Array.from({ length: N_subdiv }).map((_,i) => (i+1) / denom)\n const range = domain.map(q => ζ_H(s,q))\n domain.push(1)\n range.push(ζs)\n return { domain, range }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2227,"value":"tex`\\zeta(s,a) = \\zeta(s,a-1) - \\frac{1}{((a-1)^2)^{\\frac{s}{2}}}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2069,"value":"ζs_H_trace = {\n const denom = N_subdiv + 1\n const total = N * denom + 1\n const P = Array(total)\n\n P[0] = { x: 0, y: 0, z: 0 }\n\n // loop over each frac\n for (let i = 0; i < denom; ++i) {\n const frac = frac_parts.domain[i]\n let last = frac_parts.range[i]\n // set initial value from precomputed frac range values\n let k = i + 1\n P[k] = { x: last.re, y: last.im, z: frac }\n\n // loop from 1 to N and compute derived values\n for (let n = 1; n < N; ++n) {\n last = last.sub(ℂ(n + frac - 1).pow(2).pow(s_neg.div(2)))\n // set new point at appropriate index\n k += denom\n P[k] = { x: last.re, y: last.im, z: n + frac }\n }\n }\n \n return P\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1851,"value":"// s = ℂ(mathjs.bignumber(0.5),mathjs.bignumber('14.1347251417346937904572519835624766'))\n// s = ℂ(mathjs.bignumber(0.5),mathjs.bignumber('347.27267758442048447579709488806986'))\ns = ℂ(ℜs,ℑs)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1872,"value":"s_neg = s.neg()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1904,"value":"ζs = ζ_H(s,1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3268,"value":"ζn = (z,n) => {\n if (n == 0) return ℂ(0)\n if (n == 1) return ℂ(1)\n let cur = ℂ(1)\n for (let k = 2; k <= n; ++k) {\n cur = cur.add(ℂ(k).pow(z.neg()))\n }\n return cur\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1787,"value":"ζs_ℑs_π = ζ_H(s, Math.abs(s.im) / Math.PI) // TODO mod out by π?","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2397,"value":"ζs_q = ζ_H(s,q)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2678,"value":"tex`\\zeta(s,a+1) = \\zeta(s,a) - \\frac{1}{(a^2)^\\frac{s}{2}}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3426,"value":"ζ_a_next = (z, a, z_a) => (z_a || ζ_H(z, a)).sub(ℂ(1).div(ℂ(a).pow(2).pow(z.div(2))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2707,"value":"tex`\\zeta(s,a-1) = \\zeta(s,a) + \\frac{1}{((a-1)^2)^\\frac{s}{2}}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3437,"value":"ζ_a_prev = (z, a, z_a) => (z_a || ζ_H(z, a)).add(ℂ(1).div(ℂ(a).sub(1).pow(2).pow(z.div(2))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2683,"value":"ζs_q_next = ζs_q.sub(ℂ(1).div(ℂ(q).pow(2).pow(s.div(2))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2687,"value":"ζs_q_prev = ζs_q.add(ℂ(1).div(ℂ(q).sub(1).pow(2).pow(s.div(2))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5760,"value":"ζs_q_partials = {\n const L = [ℂ(0)]\n for (let n = 1; n <= N; ++n) {\n L[n] = L[n-1].add(ℂ(n+q).pow(s_neg))\n }\n return L\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2419,"value":"ζr = ζ_H(ℂ(1).sub(s), 1) // TODO just use reflection formula on ζs","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3092,"value":"ζs_over_ζs_reflected = ζs.div(ζr)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4072,"value":"// χ(s) χ(1-s) = 1\n// page 16 of: https://zulfahmed.files.wordpress.com/2018/08/titchmarsh-and-d-r-heath-brown-the-theory-of-the-riemann-zeta-function.pdf\n// from page 27: 1 when Re(s) = 1/2 ... maybe?\n// NB *not* xi... ξ -- which obeys ξ(s) = ξ(1-s)\ntex`\\chi(s) = \\frac{\\zeta(s)}{\\zeta(1-s)} = \\pi^{s-\\frac{1}{2}} \\frac{\\Gamma(\\frac{1-s}{2})}{\\Gamma(\\frac{s}{2})}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4070,"value":"χ = (s) => ℂ(mathjs.eval('pi^(s-1/2) gamma((1-s)/2) / gamma(s/2)', { s }))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4278,"value":"χs = χ(s)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4802,"value":"\n// https://web.stanford.edu/group/journal/cgi-bin/wordpress/wp-content/uploads/2012/09/Erickson_NatSci_2005.pdf\n// γr(s) = 1/γr(1-s)\ntex`\\gamma_{r}(s) = 2^s π^{s-1} \\sin(\\frac{πs}{2}) \\Gamma(1-s) = χ(s)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4810,"value":"γr = (s) => ℂ(mathjs.eval('2^s pi^(s-1) sin(pi s/2) gamma(1-s)', { s }))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4812,"value":"γr_s = γr(s)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5919,"value":"γr_s","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5887,"value":"tex`\\gamma_{r}(1-s) = \\frac{1}{\\chi(s)} = \\frac{\\zeta(1-s)}{\\zeta(s)} = \\frac{2 \\Gamma(s)}{(2π)^s} \\cos(\\frac{πs}{2}) = \\frac{\\Gamma(s)}{(2π)^s} (e^{\\frac{iπs}{2}} + e^{\\frac{-iπs}{2}})`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4828,"value":"γr_r = γr(s_neg.add(1))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5907,"value":"mathjs.eval('2 gamma(s) (2 pi)^-s cos(pi s/2)', {s})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5935,"value":"mathjs.eval('gamma(s) (2 pi)^-s * (exp(i*pi*s/2) + exp(-i*pi*s/2))', {s})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4024,"value":"[ ζs_over_ζs_reflected, χs, γr_s ]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5369,"value":"tex`\\zeta(s,\\frac{1}{2}) = (2^s - 1) \\zeta(s)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4273,"value":"ζs_half = ζs.mul(ℂ(2).pow(s).sub(1))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4788,"value":"// https://dlmf.nist.gov/25.11#vi\ntex`\\frac{∂}{∂a} \\zeta(s,a) = -s \\zeta(s+1,a)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4785,"value":"ζs_q_next_dq = s.neg().mul(ζ_H(s.add(1),q+1))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5440,"value":"tex`\\frac{∂^2}{∂a^2} \\zeta(s,a) = s(s+1) \\zeta(s+2,a)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5476,"value":"// TODO\n// ζs_q_next_d2q = s.mul(s.add(1)).mul(ζ_H(s.add(2),q+1))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2825,"value":"// TODO\n// ζ_approx = (t) => s","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2598,"value":"tex`\\left(\\Re{\\zeta\\left(s\\right)}+\\frac{t^{1 - \\Re{s}}}{\\sqrt{\\left(1 - \\Re{s}\\right)^2 + \\Im{s}^2}}\\cos\\left(-\\Im{s}\\ln\\left(t\\right)+\\arctan\\left(\\Im{s}, 1 - \\Re{s}\\right)\\right),\\Im{\\zeta\\left(s\\right)}+\\frac{t^{1-\\Re{s}}}{\\sqrt{\\left(1 - \\Re{s}\\right)^2 + \\Im{s}^2}}\\sin\\left(-\\Im{s}\\ln\\left(t\\right)+\\arctan\\left(\\Im{s}, 1 - \\Re{s}\\right)\\right)\\right)`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6392,"value":"γ0 = 0.5772156649015328606065120","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2432,"value":"function buildCurve(name, opts, data) {\n opts = opts || {}\n if (!data) return\n\n if (typeof opts.line === 'string') opts.line = {\n color: opts.line\n }\n \n if (opts.line && opts.line.simplify == null) opts.line.simplify = true\n \n if (typeof opts.marker === 'string') opts.marker = {\n color: opts.marker,\n size: 2,\n opacity: 0.75,\n }\n \n if (!opts.hoverinfo) opts.hoverinfo = 'skip'\n\n const modes = []\n if (opts.line) modes.push('lines')\n if (opts.marker) modes.push('markers')\n if (data.text) modes.push('text')\n\n if (!modes.length) modes.push('markers')\n return {\n name,\n type: 'scatter3d',\n mode: modes.join('+'),\n ...opts,\n ...data,\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1667,"value":"// viewof N = slider({ description: \"Total step count\", min: 0, max: 200, step: 1, value: 50 })\nN = Math.min(Math.floor(Math.abs(s.im) / π) + N_extra, 400)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6083,"value":"$32","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6503,"value":"md`\nAt zeta zeros, using a rotation factor of 1/2 leads to a beautiful double-helix zipper at ~ s.im/π -- alternates between touching zeta and hurwitz zeta partial sums... reflected partials should do something similar.\n\nThis should only be able to happen at zeta zeros -- and whether this could only happen at Re(s) = 1/2 is equivalent to RH.\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3852,"value":"// TODO is there a spiral which perfectly connects the C_n values (pendant centers)?\n// obviously clothoid-like, but could be anything (e.g. log, hyperbolic, etc.)\n// calculate endpoints, two per C_n pendant center\n\n// revisit connection between approximate zeta log spiral differences and hurwitz partials\n// this could shed light on formula for C_n with n > 0","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5413,"value":"viewof N_extra = slider({ description: \"Total steps past convergence\", min: -20, max: 50, step: 1, value: 20 })\n// N_extra = 100","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2796,"value":"viewof plot_width = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2852,"value":"viewof plot_height = html``","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6839,"value":"ζs_q_next_dq","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6265,"value":"viewof trace_config = checkbox({\n description: 'Configure traces',\n options: [\n { value: 'normalize_rotated_spirals', label: tex`ζ^•(s,t)\\ normalize` },\n { value: 'rejoin_rotoshifted_spirals', label: tex`ζ^\\tau(s,t)\\ rejoin` },\n ],\n value: []\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6086,"value":"graph2Div = html`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5663,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":411,"value":"viewof ℜs = slider({\n description: \"ℜs\",\n min: -3,\n max: 3,\n step: 0.05,\n value: 0.5\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7010,"value":"ℑs_sign = bools.includes(\"negate ℑs\") ? -1 : 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":378,"value":"ℑs = ℑs_sign * 237.224229666\n// viewof ℑs = slider({ description: 'ℑs', min: 0, max: 100, value: 22.27 })\n// viewof ℑs = slider({ description: 'ℑs', min: 0, max: 65, value: 0 })\n// viewof ℑs = slider({ description: 'ℑs', min: ρ(90).im, max: ρ(100).im, value: ρ(99).im })\n\n// ℑs = 2*π*2**7/Math.log(2)\n\n// ℑs = 14.13472514173469\n// ℑs = ρ(14).im\n// ℑs = 43*π+0.2 // 137.224229666\n// ℑs = 221.02203963877155499262847\n// ℑs = 236.524229665816205802475507955663171 // 100th zeta zero","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7005,"value":"viewof bools = Inputs.checkbox([\"negate ℑs\"])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7088,"value":"s.im","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6990,"value":"partials_folding_ratio = partials_folding_factor * Math.sign(s.im)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6892,"value":"// partials folding factor\n// viewof partials_folding_factor = Inputs.range([0, 2], { value: 2, step: 0.001 })\n\nviewof partials_folding_factor = Scrubber(linspace(0, 2, 1001), {\n loop: true,\n alternate: true,\n autoplay: false,\n delay: 40\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1491,"value":" graphDiv = html`

`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7061,"value":"q_kink = s.im / (partials_folding_ratio * π) + partials_folding_ratio\n// viewof q_kink = Inputs.range([40, 100], { value: 45, step: 0.01 })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5634,"value":"viewof rotoshift_partial_min = slider({\n min: 0,\n max: 20,\n step: 0.125,\n value: 2.5,\n description: \"max spool to show for rotoshifted traces\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4333,"value":"viewof trace_visibility = checkbox({\n description: \"Show or hide specific sets of traces\",\n options: [\n { value: \"hurwitz_spirals\", label: tex`ζ(s,t)` },\n { value: \"fixed_half_hurwitz_spirals\", label: tex`ζ_θ[*](s,t)` },\n { value: \"rotated_hurwitz_spirals\", label: tex`ζ_θ(s,t)` },\n { value: \"rotoshifted_spirals\", label: tex`ζ^{\\tau*}(s,t)` },\n { value: \"rotoshifted_spirals_reversed\", label: tex`ζ(s) - ζ^\\tau(s,t)` },\n { value: \"rescaled_zeta_partials\", label: tex`ζn(1-s)\\ rescaled` },\n { value: \"approx_spirals\", label: tex`r(θ) ≈ ζn` },\n { value: \"harmonic_spirals\", label: tex`ζ(s,Hn) ≈ r(θ)` }\n ],\n value: [\"rotated_hurwitz_spirals\", \"rotoshifted_spirals\", \"rotoshifted_spirals_reversed\"]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5807,"value":"q","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":931,"value":"// viewof q = html``\nq = 128.5\n\n// TODO formula for second derivative... represents change in curvature?\n// also, find points where second derivative equals 0...\n// represents point where osculating circle radius goes to infinity?\n// this point would represent the origin for any euler-spiral-based formula","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":822,"value":"viewof r_scale = slider({ min: 0.01, max: 20, step: 0.01, value: 2, description: 'complex plane size' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4133,"value":"viewof height_aspect_scale = slider({ min: 0.1, max: 10, step: 0.1, value: 2, description: 'height aspect scale factor' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":337,"value":"viewof N_subdiv = slider({ description: \"Step subdivisions\", min: 0, max: 50, step: 1, value: 19 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4152,"value":"viewof width_aspect_scale = slider({ min: 0.1, max: 10, step: 0.1, value: 1.5, description: 'width aspect scale factor' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6616,"value":"// TODO fix z offsets, in particular: q hurwitz partials (olive) and derivative vectors","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1522,"value":"camera_current = cameras.front","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4873,"value":"C_n_list = {\n const list = [];\n const halfwinds = Math.floor(\n (Math.abs(s.im) * Math.log(Math.abs(s.im) / π)) / π\n );\n const centers = 2; // Math.floor((halfwinds - 1) / 2) // TODO: +1 instead of -1?\n for (let i = 0; i < centers; ++i) {\n list[i] = ζ_H(s, Math.abs(s.im) / ((2 * i + 1) * π));\n }\n return list;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4968,"value":"C_0_rot = C_n_list[0]\n .sub(ℂ(Math.abs(s.im) / π).pow(s_neg))\n .mul(\n mathjs.exp(\n ℂ(\"i\").mul(π * (Math.abs(s.im) / π - Math.floor(Math.abs(s.im) / π)))\n )\n )\n .arg() - C_n_list[0].sub(ℂ(Math.abs(s.im) / π).pow(s_neg)).arg()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4849,"value":"ζs_rotoshift_smooth = (t, z_t) => {\n const ζs_t_origin = z_t || ζ_H(s, t);\n\n let d = ℂ(t).pow(s_neg);\n d = ζs_t_origin.sub(d);\n\n let xn = Math.abs(s.im) / (2 * π * t) - 1 / 2; // = (Math.abs(s.im) - π*t) / (2*π*t)\n\n // just use ζ(s,t) - t^-s if past C_0 center\n if (xn <= 0) return d;\n\n const rotation = mathjs.exp(ℂ(\"i\").mul(π * t * partials_folding_ratio));\n\n let k = -1;\n while (++k < xn) {\n const shift = k > 0 ? ζn(s_neg.add(1), k).mul(χs) : ℂ(0);\n\n // shift down to 0 to rotate\n d = d.sub(shift);\n\n d = d.mul(rotation);\n\n // unshift back to position\n d = d.add(shift);\n }\n\n // if rejoin rotoshifted requested, flip over line of symmetry\n if (trace_config.includes(\"rejoin_rotoshifted_spirals\")) {\n // for (let k = 0, len = Math.min(C_n_list.length, xn); k < len; ++k) {\n d = d\n .mul(mathjs.exp(ℂ(\"i\").mul(-C0_lower.arg())))\n .conjugate()\n .mul(mathjs.exp(ℂ(\"i\").mul(C0_upper.arg())));\n // }\n }\n\n return d;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3712,"value":"$34","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4236,"value":"$35","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4164,"value":"{\n const ζs_unit_frac = ℂ(ζs_H_trace[1].x, ζs_H_trace[1].y)\n const ζs_neg_unit_frac = ℂ(ζs_H_trace[N_subdiv].x, ζs_H_trace[N_subdiv].y)\n // return ζs_half.sub(ζs_neg_unit_frac).abs()\n // return ζs_neg_unit_frac.sub(ζs).abs()\n return ζs_unit_frac.abs()\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2294,"value":"rval = r_scale // * ζs.abs()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2747,"value":"viewof reflected_rotoscale = slider({ min: -1, max: 1, step: 0.01, value: 1, description: 'reflected zeta rotation/rescale factor' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1636,"value":"viewof z_axis_log_mode = checkbox({\n description: 'Show z-axis in log mode?',\n options: [{ value: true, label: 'log mode' }],\n value: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2281,"value":"viewof z_axis_to_zero = checkbox({\n description: 'Exclude negative unit on z-axis?',\n options: [{ value: true, label: tex`min_z : 0` }],\n value: false\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1562,"value":"camera_current","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3940,"value":"ζs_rotoshift_alt = (t, z_t) => {\n const ζs_t_origin = z_t || ζ_H(s, t)\n\n const scalefrac = t - Math.floor(t)\n // rotate all sums by e^(2π * frac(t))\n // NB no longer the above ^^ -- but slight derivation\n const rotation = mathjs.exp(ℂ('i').mul(π * scalefrac * partials_folding_ratio))\n\n let d = ℂ(t).pow(s_neg)\n d = ζs_t_origin.sub(d).mul(rotation)\n\n let xn = Math.abs(s.im) / (2 * π * (t + 1/2) - π) - 1/2\n // if (xn < 0) ... TODO turn around... appears we're not taking shortest path here\n\n let k = 0\n while (++k < xn) {\n // shift one normalized/reversed partial step, rotate, shift back\n const shift = ζn(s_neg.add(1), k).mul(ζs_over_ζs_reflected)\n // rotate by difference between \n d = d.sub(shift).mul(rotation).add(shift)\n }\n\n return d\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3724,"value":"ζs_q_rotoshift = ζs_rotoshift_smooth(q)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3970,"value":"ζn(s_neg.add(1), 3).mul(ζs).div(ζr).toString()","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5346,"value":"Cn_upper = (n) => C_n_list[n].sub(ℂ(Math.abs(s.im) / ((2*n+1)*π)).pow(s_neg)) // TODO we need a sum from 0 to n","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5361,"value":"Cn_lower = (n) => Cn_upper(n).mul(mathjs.exp(ℂ('i').mul(π * Math.abs(s.im) / ((2*n + 1)*π))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5217,"value":"C0_upper = C_n_list[0].sub(ℂ(Math.abs(s.im) / π).pow(s_neg))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5222,"value":"C0_lower = C0_upper.mul(mathjs.exp(ℂ(\"i\").mul(Math.abs(s.im))))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3855,"value":"$36","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3159,"value":"viewof rotoshift_num = slider({ min: 14, max: 28, value: 1, description: 'rotoshift numerator' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2819,"value":"viewof rotoshift_denom = slider({ min: 0.3, max: 20, step: 0.1, value: 1, description: 'rotoshift denominator' })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2868,"value":"$37","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5166,"value":"ζs_q_rotoshift_dq = mathjs.exp(ℂ('i').mul(π*q*partials_folding_ratio)).mul(\n ℂ('i').mul(π*partials_folding_ratio).mul(ζs_q_next).add(ζs_q_next_dq)\n)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5436,"value":"$38","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5606,"value":"binom(0,1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5573,"value":"binom = (n,k) => n >= k ? mathjs.eval('combinations(n,k)', { n,k }) : n === 0 ? (k === 0 ? 1 : 0) : mathjs.eval('(-1)^k * combinations(n+k-1,k)', { n: -n, k })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5541,"value":"mathjs.eval('s * (s+1)', {s})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5533,"value":"ζs_q_rotoshift_d2q = mathjs.exp(ℂ('i').mul(π*q*partials_folding_ratio)).mul(\n ℂ('i').mul(π*partials_folding_ratio).mul(ζs_q_next).add(ζs_q_next_dq)\n)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3221,"value":"$39","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2589,"value":"$3a","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2431,"value":"$3b","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2548,"value":"$3c","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2630,"value":"axis_traces = [\n {\n name: '|ℂ|',\n x: [0,1,rval],\n y: [0,0,0],\n z: [0,0,0],\n type: 'scatter3d',\n mode: 'lines+markers',\n line: {\n color: 'rgba(0,0,0,0.6)',\n },\n marker: {\n size: 2,\n // opacity: 0.6,\n },\n },\n {\n name: 'ℑ{s}',\n x: [0,0],\n y: [0,0],\n z: [z_range_min,z_range_max],\n type: 'scatter3d',\n mode: 'lines',\n line: {\n color: 'rgba(0,0,0,0.6)',\n dash: 'dot',\n },\n // visible: 'legendonly',\n },\n]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":862,"value":"$3d","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1190,"value":"cameras = {\n return {\n top: {\n center: {\n x: 0,\n y: 0,\n z: 0,\n }, \n eye: { \n x: 0,\n y: 0,\n z: 1000,\n }, \n up: {\n x: 1,\n y: 0,\n z: 1,\n },\n projection: {\n type: 'orthographic',\n },\n },\n front: {\n center: {\n x: 0,\n y: 0,\n z: 0,\n }, \n eye: { \n x: 0,\n y: 1,\n z: 0,\n }, \n up: {\n x: 0,\n y: 0,\n z: 0,\n },\n projection: {\n type: 'orthographic',\n },\n },\n front_diagonal: {\n center: {\n x: 0,\n y: 0,\n z: 0,\n }, \n eye: { \n x: -1,\n y: -1,\n z: 1,\n }, \n up: {\n x: 0,\n y: 0,\n z: 1,\n },\n projection: {\n type: 'orthographic',\n },\n },\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1263,"value":"size = Math.min(width, 600)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2904,"value":"z_range_max = z_axis_log_mode ? Math.sign(N+1)*Math.log10(Math.abs(N+1)) : N+1","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2907,"value":"z_range_min = -(1-Number(z_axis_to_zero))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1367,"value":"$3e","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":864,"value":"{\n Plotly.react(\n graphDiv,\n traces,\n layout,\n {\n scrollZoom: true,\n displayModeBar: true,\n displaylogo: false,\n // modeBarButtonsToRemove: ['toImage'],\n // responsive: true,\n },\n )\n \n graphDiv.removeAllListeners()\n\n // graphDiv.on('plotly_relayout', (event) => {\n // set_camera(event['scene.camera'])\n // })\n\n graphDiv.on('plotly_click', (event) => {\n console.log('click', event)\n })\n\n graphDiv.on('plotly_doubleclick', (event) => {\n console.log('doubleclick', event)\n })\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3396,"value":"---\n## Notes\n\n\n### 2-adic Hurwitz calculation\n\ncan you make a similar triangles argument for the triangle \"bumped out\" of the unit segment by ζ(s,1/2)?\n* should be identical to the triangle bumped out by ζ(s,3/4) from ζ(s) to ζ(s,1/2) segment\n * we know this value exactly when we know ζ(s,1/2), as it's just ζ(s,3/2)\n* this triangle can be projected onto ζ(s,1):ζ(s,1/2) base\n * third point should be ζ(s,3/4)\n * knowing ζ(s,3/4) gets you ζ(s,1/4)\n * should this be the same triangle when divisions is 8? e.g. ζ(s,5/8) for ζ(s,4/8):ζ(s,6/8)\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":324,"value":"md`---\n## Dependencies\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4850,"value":"π = Math.PI","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":155,"value":"Plotly = require(\"https://cdn.plot.ly/plotly-latest.min.js\")","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2456,"value":"linspace = {\n const m = await import(\"https://cdn.skypack.dev/linspace@1.0.2?min\");\n return m.default;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6913,"value":"import { Scrubber } from \"@mbostock/scrubber\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":409,"value":"import {slider, checkbox, number} from \"@jashkenas/inputs\"","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1818,"value":"import { ζ_H_, mathjs } from '2a0dd2a4796acc42'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5741,"value":"ζ_H = (z,a) => {\n if (a > 1e6) return ℂ(NaN)\n return ζ_H_(z,a)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2022,"value":"ℂ = mathjs.complex","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2879,"value":"ℚ = mathjs.fraction","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6525,"value":"ρ = (n) => ℂ(0.5,ρ_i[n-1])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6522,"value":"$3f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W40","stdlib":"1","titling":"auto"}} 31:"$30:body"