E-009a — Suzuki screw function g(t): implementation + eq. 2.2 validation + tabulation

08-17-2026 · 10001 rows · script research/numerics/scripts/E-009-suzuki-screw-function.py

Metadata

mpmath implementation of Suzuki's explicit screw function g(t) (arXiv:2606.09096v1 eq. 1.3); validated against eq. 2.2 near-origin expansion; tabulated on [0,10] step 1e-3 for E-009b Galerkin

Validation checks

A1_lerch_convention	PASS
A2_evenness	PASS
A3_A_limit	PASS
A4_r_curvature_converges	PASS
B_table_finite	PASS
B_spot_check_vs_mpmath	PASS

Key values

A_eq2.2	`0.707546365655439`
c2_lim_r_over_t2	`-0.8750599754042412`
digamma_coeff_-half(psi(1/4)-log pi)	`2.6860917096128327`
Phi(1,2,1/4)	`17.19732915450711`
g(log2)	`-0.06355558603756917`
g(1)	`-0.0440073052368517`
g(5)	`-0.045371349558830154`
g(10)	`-0.06209008245757719`

Parameters

t_max	`10.0`
step	`0.001`
validation_dps	`30`
n_sieve_max	`22028`

Caveats

float64 table: Lerch combination has cancellation for small t (~4 digits lost at t=1e-3); spot-checked vs mpmath to 1e-9
prime sum is piecewise linear in t (kinks at log n) — exact via cumsum, no interpolation error
g even: only t >= 0 tabulated

Chart

Data preview (10001 rows total)

t	g_t	prime_sum_t
0.0000	0.000000000000e+00	0.000000000000e+00
0.0010	-2.747206277331e-03	0.000000000000e+00
0.0020	-4.803015395265e-03	0.000000000000e+00
0.0030	-6.598950484153e-03	0.000000000000e+00
0.0040	-8.226736601324e-03	0.000000000000e+00
0.0050	-9.729937036770e-03	0.000000000000e+00
0.0060	-1.113421001570e-02	0.000000000000e+00
0.0070	-1.245650964240e-02	0.000000000000e+00
0.0080	-1.370888589774e-02	0.000000000000e+00
0.0090	-1.490034855153e-02	0.000000000000e+00
0.0100	-1.603789097998e-02	0.000000000000e+00
0.0110	-1.712709997647e-02	0.000000000000e+00
0.0120	-1.817254187329e-02	0.000000000000e+00
0.0130	-1.917801903422e-02	0.000000000000e+00
0.0140	-2.014674695453e-02	0.000000000000e+00
0.0150	-2.108148050452e-02	0.000000000000e+00
0.0160	-2.198460636164e-02	0.000000000000e+00
0.0170	-2.285821224724e-02	0.000000000000e+00
0.0180	-2.370413981830e-02	0.000000000000e+00
0.0190	-2.452402577109e-02	0.000000000000e+00
0.0200	-2.531933426886e-02	0.000000000000e+00
0.0210	-2.609138286873e-02	0.000000000000e+00
0.0220	-2.684136349989e-02	0.000000000000e+00
0.0230	-2.757035962015e-02	0.000000000000e+00
0.0240	-2.827936038365e-02	0.000000000000e+00