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# Independent contour enclosure for T692
This note derives the explicit error bounds used by checks.py. Its analytic
starting point is Akatsuka–Murakami, arXiv:2402.14216v1, equations (3.1)–(3.5):
https://arxiv.org/html/2402.14216v1#S3.
Write s=3/2+it and F(s)=zeta(s)zeta(1-s)/sin(pi s). Parametrizing the upward
vertical contour by t, and using conjugate symmetry, gives
g_0 = -log(2 pi) + gamma + 2 Re integral_0^infinity F(3/2+it) dt,
g_n = 1/n + 2(-1)^n Re integral_0^infinity
F(3/2+it) binom(3/2+it,n) dt, n>=1.
The positive sign after 1/n agrees with AM's correction to the older formula.
The gamma ratio in that formula is evaluated as the finite polynomial
(-1)^n binom(s,n), avoiding unnecessary gamma evaluation and its poles.
The functional equation gives
zeta(1-s) = 2 (2 pi)^(-s) Gamma(s) cos(pi s/2) zeta(s).
For real t, Gamma reflection and recurrence give
|Gamma(3/2+it)|^2 = pi (t^2+1/4)/cosh(pi t),
|cos(pi(3/2+it)/2)|^2 = cosh(pi t)/2,
|sin(pi(3/2+it))| = cosh(pi t).
Since |zeta(3/2+it)| <= zeta(3/2) < 1+integral_1^infinity u^(-3/2)du = 3,
these yield
|F(3/2+it)| <= 9/(2 pi) sqrt(t^2+1/4)/cosh(pi t).
Cauchy's coefficient formula for (1+w)^s on circles of radius r<1, letting
r increase to 1, gives the AM Section 3 bound
|binom(3/2+it,n)| <= 2^(3/2) exp(pi |t|/2).
For t>=0, use 1/cosh(pi t)<=2 exp(-pi t) and
sqrt(t^2+1/4)<=t+1/2. Including both conjugate tails, the absolute omitted
contribution to every g_n, including n=0, is at most
(18*sqrt(8)/pi) integral_T^infinity (t+1/2) exp(-pi t/2) dt
< 18 integral_T^infinity (t+1/2) exp(-3t/2) dt
= (12T+14) exp(-3T/2).
The checker uses T=60, so this uniform tail bound is below 6.015e-37. It
integrates the finite interval with Arb at 180 bits and absolute and relative
tolerances 2^(-140), then adds the tail as an error ball. The integrand is
meromorphic; any integration enclosure reaching a pole becomes nonfinite and
causes subdivision. There is no branch cut or real-part operation inside the
complex analytic callback. The real part is taken after integration.
The same bound at T=0 is 14. Thus |g_n|<15 for n>=1. For N=100 and |z|<1,
we may safely bound the omitted Taylor series of pi g(1+z) by
16 |z|^(N+1)/(1-|z|).
This is added when comparing the 101-term polynomial with the two finite
cotangent sums defining pi g(x). It is a uniform proved tail bound, not an
estimate from observed coefficients.
Controls run before the general comparison: Arb integrates t over [0,1] and
must enclose 1/2 tightly; the contour results must recover g_0=-1 and g_1=1/2.
Every enclosure is tested for finiteness and width. A perturbation of 10^(-30)
must fail to overlap the recorded coefficient, well outside the contour radius.
The archived skill manifest is
../20261109T150707Z-build-skills/manifest.json, copied to the draft attachment
skill-manifest.json. Both archived local and live skill SHA-256 values are
886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b.