[{"type":"thinking","thinking":"mpmath.stieltjes is fast (0.15s at k=500). Question: accuracy for large k? Stieltjes constants grow in magnitude and are hard to compute accurately for large k; mpmath's implementation (numerical integration) may lose accuracy for k > ~1000 at fixed dps. Test accuracy at k=500, 1000, 2000 against known values... Known: γ_1000 etc. — I don't remember exact values, but we can validate the END PRODUCT: λ_n vs E-008's certified zero-side values for n ≤ 116, λ_1 exact, and asymptotic trend. Also positivity. Also cross-validate mpmath stieltjes vs Arb? Arb's acb zeta has no stieltjes... Alternative validation: compute λ_n via a completely different route at sample n (e.g., zero-side with rigorous tail for small n — E-008 did exactly that to n=116).\n\nPrecision plan: working dps ~ ceil(0.31·n_max) + 60. For n_max = 2000: ~680 dps. mpmath stieltjes at high dps for k up to 2000 — check accuracy: compare stieltjes(k) at dps 100 vs dps 200 for k=500,1000,2000 — agreement digits estimate error.\n\nAlso need polygamma at 1/2: ψ^{(m)}(1/2) = (−1)^{m+1} m! (2^{m+1}−1) ζ(m+1) — for large m, ζ(m+1) → 1; mpmath handles.\n\nAnd log(tζ(1+t)) series recursion: tζ(1+t) = Σ_{m≥0} c_m t^m with c_0 = 1, c_m = (−1)^{m−1}γ_{m−1}/(m−1)! for m ≥ 1. log: η_j^ζ := [t^j] log Σ c_m t^m: standard recursion: L_j = (c_j − (1/j)Σ_{i=1}^{j−1} i L_i c_{j−i})·... for log S where S = Σc_m t^m, L = log S: j L_j = j c_j − Σ_{i=1}^{j−1} i L_i c_{j−i} ... (S'/S = L': Σ j L_j t^{j−1} = S'/S ⇒ jL_j = j c_j − Σ_{i=1}^{j−1} iL_i c_{j−i}? Derive: L'·S = S': (Σ_{j≥1} jL_j t^{j−1})(Σ c_m t^m) = Σ_{m≥1} m c_m t^{m−1}. Coefficient of t^{j−1}: Σ_{i=1}^{j} i L_i c_{j−i} = j c_j ⇒ jL_j = j c_j − Σ_{i=1}^{j−1} iL_i c_{j−i} ✔ (c_0 = 1).\n\nThen η_j = (−1)^{j+1}/j + L_j − (log π/2)[j==1] + (1/2^j)ψ^{(j−1)}(1/2)/j! and λ_n = Σ_{j=1}^n C(n,j)·j·η_j.\n\nWait — check the log(s) part: log s at s=1+t: log(1+t) = Σ (−1)^{j+1}t^j/j ✔. The ½ factor: ξ = ½s(s−1)π^{−s/2}Γ(s/2)ζ(s): log ξ = −log 2 + log s + log(s−1) − (s/2)logπ + logΓ(s/2) + log ζ(s). log(s−1) + log ζ = log((s−1)ζ(s)) = log(tζ(1+t)) ✔. −(s/2)logπ contributes −(logπ)/2·t ⇒ −(logπ)/2 to j=1 coeff ✔. logΓ(s/2) = logΓ(1/2 + t/2): coefficients (1/j!)(1/2^j)ψ^{(j−1)}(1/2) ✔.\n\nSanity check λ_1: η_1 = 1 + L_1 − (logπ)/2 + ½ψ(1/2); L_1 = c_1 = γ_0. ψ(1/2) = −γ_E − 2log 2. η_1 = 1 + γ_E − (logπ)/2 − γ_E/2 − log 2 = 1 + γ_E/2 − log 2 − (log π)/2 = 1 + γ_E/2 − ½log(4π) ✔ matches earlier. λ_1 = C(1,1)·1·η_1 = η_1 = 0.0230957... ✔.\n\nValidation plan for E-011:\n1. λ_1 vs known 0.02309570896612103381 (30+ digits).\n2. λ_n for n=2..116 vs E-008 zero-side values (within their tail bounds).\n3. λ_n/(n/2 (log n − log(2π) + γ_E − 1)) trend → 1 (Arias de Reyna).\n4. All λ_n > 0.\n5. Self-consistency: recompute at 1.5× precision and compare (accuracy estimate).\n\nScope: n_max = 2000 first (dps ~ 700), then extend: n = 10^4 needs dps ~ 3100 — mpmath stieltjes at k~10^4, dps 3100: cost? stieltjes(k) each ~0.15s at dps 50... at dps 3100 slower; and we need ALL k up to n. Let me measure scaling in the run. Also the λ_n summation: C(n,j) j η_j with cancellation — need the precision budget.\n\nNote: mpmath.stieltjes(k) for large k: known accuracy issues? mpmath docs say stieltjes uses ... it computes via the formula involving zeta derivatives... Let me just test empirically: compute at dps 100 vs 150 for k = 1000, 2000 and count agreeing digits.\n\nAlso — IMPORTANT alternative check: E-008's zero-side λ_n for n ≤ 116 are CERTIFIED (tail bound < 5%). And the zero side to n=2000 computed from zeros has large tail bounds beyond 116 but still gives an approximate target (few-% accuracy up to n~300?). Fine.\n\nPlan E-011 script:\n- params: n_max = 2000, dps = 700 (initial), maybe extend flag --n-max.\n- compute stieltjes γ_0..γ_{n_max−1} (cache in CSV/JSON? per protocol, data cached — the run caches its own output CSV; also save intermediate stieltjes? not necessary).\n- compute L_j recursion at dps, η_j, λ_n.\n- self-consistency check: recompute every-10th λ at dps·1.4 and compare.\n- validations as above.\n- outputs CSV: n, lambda_n, lambda_n/(n/2 log n), ratio to refined asymptotic, plus E-008 comparison for n≤116.\n\nRuntime: stieltjes calls: n_max=2000 at dps 700 — each maybe 0.5-2s ⇒ 2000 × 1s = 33 min. Hmm, too slow. Check timing at dps 700 first. If slow, reduce n_max to 1000 or optimize (stieltjes in mpmath is O(?)-per-call; maybe batch? There's no batch API). Alternatively compute Stieltjes via Arb acb zeta jet at... hmm, Arb CAN compute ζ jet at s=1? The C library acb_dirichlet_zeta_jet(s, len, deflate=1) handles the pole via deflation (returns jet of (s−1)ζ). But python-flint doesn't wrap acb_dirichlet. \n\nAlternative: use acb_series at s₀ = 1 + ε?? no.\n\nHmm — actually, wait: maybe I can get the ξ jet directly: acb_series has 'lgamma', 'gamma', 'zeta'... The zeta method failed at the pole. But what about evaluating zeta series at s = 2 + t (regular!) and then... no.\n\nAlternatively use mpmath for the whole thing but with a FASTER stieltjes: mpmath's stieltjes implementation: for n ≤ some limit uses precomputed? It's fast. Measure at dps 700.","signature":""},{"type":"text","text":"mpmath has fast Stieltjes constants. Checking timing/accuracy at working precision (n=2000 needs ~700 dps)."},{"type":"tool_use","id":"Bash_144","name":"Bash","input":{"command":"cd D:/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport mpmath as mp, time\nfor dps in (100, 300, 700):\n    mp.dps = dps\n    t0=time.time(); mp.stieltjes(1000); t1=time.time()-t0\n    t0=time.time(); mp.stieltjes(2000); t2=time.time()-t0\n    print(f'dps={dps}: stieltjes(1000) {t1:.2f}s, stieltjes(2000) {t2:.2f}s')\n# accuracy: compare across precisions\nmp.dps = 300; a = mp.stieltjes(1000)\nmp.dps = 200; b = mp.stieltjes(1000)\nmp.dps = 250; diff = abs(mp.mpf(a)-mp.mpf(b))\nprint('g1000 dps300 vs dps200 rel diff:', mp.nstr(diff/abs(mp.mpf(a)), 3))\n\"","description":"Time and accuracy-check mpmath stieltjes at high precision","timeout":300000},"extra_content":null}]