E-015 — Li coefficients arithmetic side n<=3000 (Bombieri-Lagarias Thm 2, zero-free)
08-18-2026 · 3000 rows · script
research/numerics/scripts/E-015-li-coefficients-n3000.py
Metadata
Bombieri-Lagarias Theorem 2 (eta-form): lambda_n from Stieltjes constants (parallel mpmath sweep, series division) + Arb zeta values; zero-free
Validation checks
| V1_lambda1_exact | PASS |
| V2_matches_E008_zero_side | PASS |
| V3_all_positive | PASS |
| V5_precision_self_consistent | PASS |
Key values
| lambda_1 | 0.0230957089661210338143102479065 |
| lambda_10 | 2.27933936319316 |
| lambda_100 | 118.603775376791 |
| lambda_1000 | 2326.05316168647 |
| lambda_n_max | 8617.21920730109 |
| asymptotic_ratio_at_n_max | 0.99984452 |
| self_consistency_rel_dev | 3.81e-1005 |
Parameters
| n_max | 3000 |
| dps | 1450 |
| arb_bits | 5029 |
Caveats
- mpmath 1.3 + Python 3.14: mp.zeta broken beyond ~16 digits; zeta values via flint.arb; mp.zetazero unaffected
- mpmath arithmetic is not interval-rigorous; certification rests on self-consistency + E-008 cross-checks; an Arb ball-arithmetic upgrade would make this fully rigorous
Chart
Data preview (3000 rows total)
| n | lambda_n | lambda_over_refined_asymptotic |
|---|---|---|
| 1 | 0.02309570896612103381431025 | -0.020432700758 |
| 2 | 0.09234573522804667038572849 | -0.058912215273 |
| 3 | 0.207638920554324803791492 | -0.119122286233 |
| 4 | 0.3687904794922416385905115 | -0.210889970949 |
| 5 | 0.5755427144611774524311064 | -0.353514714452 |
| 6 | 0.8275660122823792974250028 | -0.588300705195 |
| 7 | 1.12446011757095949058282 | -1.02072451437 |
| 8 | 1.465755677147060632655515 | -2.02206821942 |
| 9 | 1.850916048382534155326045 | -6.48384724156 |
| 10 | 2.279339363193157743693034 | 10.8737531567 |
| 11 | 2.750360838220196060354547 | 3.64389346629 |
| 12 | 3.263255320624619848079086 | 2.4253619238 |
| 13 | 3.817240057847945987104368 | 1.92997353236 |
| 14 | 4.411477678680598512080641 | 1.66548065172 |
| 15 | 5.045079372026793458535111 | 1.50356300876 |
| 16 | 5.717108248868792639419067 | 1.39597654231 |
| 17 | 6.426582872117202901145541 | 1.32052398569 |
| 18 | 7.172480938291722959252971 | 1.26556966769 |
| 19 | 7.953743094311900304808225 | 1.22442756614 |
| 20 | 8.769276872093215129599461 | 1.19298386123 |
| 21 | 9.617960723142918982361396 | 1.16856970436 |
| 22 | 10.49864813493487469708756 | 1.14937901773 |
| 23 | 11.41017181051510084091179 | 1.13414681838 |
| 24 | 12.35134789255131701703406 | 1.12196150599 |
| 25 | 13.32098021289157579994336 | 1.11215025682 |