← Session 5 — Phase 0: paper deep-read + Bombieri–Lagarias literature
thinking (1745 chars)
acb_series HAS `zeta`! So I can build the series zeta(1+t) — Arb's acb_series zeta... which wraps acb_dirichlet? Let me test: create acb_series with value at point... python-flint acb_series: flint.acb_series([coeffs], prec=?). How to get zeta(1+t) series: maybe `flint.acb_series([1, 1], n).zeta()`? i.e., zeta of the series (1 + t) — series composition evaluating ζ at a series point with pole — Arb handles Laurent? acb_series.zeta() probably computes ζ(s₀ + t)-jet if s₀ is the constant coeff... risky with pole at 1+t's constant = 1. Test directly: does .zeta() on series with constant 1 work (Laurent with valuation −1)?
Test now. If it works: ξ jet: xi_series = 0.5*(1+t)*(t)* π^{−(1+t)/2} * gamma((1+t)/2) * zeta(1+t) — the t·zeta(1+t) product — if zeta series has Laurent valuation −1, series arithmetic handles it. Then log ξ series → η_j coeffs → λ_n = Σ C(n,j) j η_j. All in ball arithmetic at precision ~ 0.31n + 60 digits.
Then E-011: compute λ_n for n up to... start: n ≤ 2000 (match E-008), validate: vs E-008 zero-side (n ≤ 116), vs λ_1 exact, vs Arias de Reyna asymptotic (n/2)(log n − log(2π) + γ − 1)... note E-008 used (n/2)log n as leading; the refined asymptotic includes the constant: λ_n ~ (n/2)(log n − log(2π) + γ_E − 1)? The note says λₙ ~ (n/2)(log n − log(2π) + γ − 1). Verify with our data.
Then push to n = 10^4 (and maybe 10^5 if fast). Precision: n·log10(2) + 50 digits; for n=10^4: 3060 digits. Arb jet of length n at that precision: cost? acb_series ops are polynomial multiplications (FFT) — n=10^4 length at 3000 digits: heavy but OK (~seconds-minutes). n=10^5: 30k digits × 10^5 length — maybe minutes-hours. Start with n_max = 10^4, and a single spot at n = 10^5 if feasible.
Test the series zeta now.
`acb_series.zeta` exists. Testing whether it handles the Laurent series at s=1.
tool: Bash
input
{
"command": "cd D:/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport flint\nflint.ctx.prec = 200\nt = flint.acb_series([1, 1], 8) # series 1 + t\nz = t.zeta()\nprint('zeta(1+t) valuation:', z.valuation(), 'length:', z.length())\nprint(z)\n\"",
"description": "Test acb_series zeta at the pole s=1"
}
raw content_json (database row)
[{"type":"thinking","thinking":"acb_series HAS `zeta`! So I can build the series zeta(1+t) — Arb's acb_series zeta... which wraps acb_dirichlet? Let me test: create acb_series with value at point... python-flint acb_series: flint.acb_series([coeffs], prec=?). How to get zeta(1+t) series: maybe `flint.acb_series([1, 1], n).zeta()`? i.e., zeta of the series (1 + t) — series composition evaluating ζ at a series point with pole — Arb handles Laurent? acb_series.zeta() probably computes ζ(s₀ + t)-jet if s₀ is the constant coeff... risky with pole at 1+t's constant = 1. Test directly: does .zeta() on series with constant 1 work (Laurent with valuation −1)?\n\nTest now. If it works: ξ jet: xi_series = 0.5*(1+t)*(t)* π^{−(1+t)/2} * gamma((1+t)/2) * zeta(1+t) — the t·zeta(1+t) product — if zeta series has Laurent valuation −1, series arithmetic handles it. Then log ξ series → η_j coeffs → λ_n = Σ C(n,j) j η_j. All in ball arithmetic at precision ~ 0.31n + 60 digits.\n\nThen E-011: compute λ_n for n up to... start: n ≤ 2000 (match E-008), validate: vs E-008 zero-side (n ≤ 116), vs λ_1 exact, vs Arias de Reyna asymptotic (n/2)(log n − log(2π) + γ − 1)... note E-008 used (n/2)log n as leading; the refined asymptotic includes the constant: λ_n ~ (n/2)(log n − log(2π) + γ_E − 1)? The note says λₙ ~ (n/2)(log n − log(2π) + γ − 1). Verify with our data.\n\nThen push to n = 10^4 (and maybe 10^5 if fast). Precision: n·log10(2) + 50 digits; for n=10^4: 3060 digits. Arb jet of length n at that precision: cost? acb_series ops are polynomial multiplications (FFT) — n=10^4 length at 3000 digits: heavy but OK (~seconds-minutes). n=10^5: 30k digits × 10^5 length — maybe minutes-hours. Start with n_max = 10^4, and a single spot at n = 10^5 if feasible.\n\nTest the series zeta now.","signature":""},{"type":"text","text":"`acb_series.zeta` exists. Testing whether it handles the Laurent series at s=1."},{"type":"tool_use","id":"Bash_142","name":"Bash","input":{"command":"cd D:/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport flint\nflint.ctx.prec = 200\nt = flint.acb_series([1, 1], 8) # series 1 + t\nz = t.zeta()\nprint('zeta(1+t) valuation:', z.valuation(), 'length:', z.length())\nprint(z)\n\"","description":"Test acb_series zeta at the pole s=1"},"extra_content":null}]