back to table · edit · history · where entries came from · files · download
3101 bytes, as of the version from 2026-10-29 00:57 (current). Recorded here, not run.
# Independent continuation used to check T595
The generator uses the finite formulas in Cunha and Freitas,
https://arxiv.org/html/2404.12114v1, Theorems 2.7 and 2.8. Its integer
coefficients follow the signed Stirling recurrence; its even-dimensional
rational term uses the polynomial in Lemma A.4 and normalization (A.6).
All production transcendental operations are Arb ball operations.
The check instead starts from the spectrum and continues the tail directly.
For d >= 2 put a=(d-1)/2 and x=l+a. The eigenvalue is x^2-a^2 and the
multiplicity is the exact polynomial
P(x) = 2x/(d-1)! product_{j=1}^{d-2}(x+j-a) = sum_j c_j x^j.
Split off l=1,...,N-1, with N=max(30,2d), and put q=N+a. Expanding the remaining
factor (1-a^2/x^2)^(-s) gives the tail zeta function
sum_j c_j sum_{k>=0} (s)_k/k! a^(2k) zeta(2s+2k-j,q).
The expansion is initially justified in the absolute-convergence half-plane.
Near s=0 the large-k series and its derivative converge normally since a/q<1;
the finite small-k terms are meromorphic, yielding analytic continuation.
For k=0, the derivative contribution is 2 zeta'(-j,q). For k>=1 and
2k-j != 1, it is a^(2k) zeta(2k-j,q)/k. At the pole 2k-j=1 use
(s)_k/k! = (s/k)(1+H_(k-1)s+O(s^2)),
zeta(1+2s,q) = 1/(2s)-psi(q)+O(s),
so the derivative contribution is
a^(2k)/k [H_(k-1)/2-psi(q)].
The determinant logarithm is the finite sum of m_l log(lambda_l) minus this
derivative. checks.py constructs P independently with rational coefficients;
it does not use Stirling numbers or the generator's coefficient functions to
compute this independent value. mpmath evaluates its Hurwitz functions at
180 decimal digits. Positive Hurwitz zeta evaluations receive an additional
ceil(p log10(q))+20 digits: mpmath 1.4.1 uses an absolute stopping tolerance,
which otherwise loses relative accuracy for tiny zeta(p,q) before multiplication
by a^(2k). The run isolated this effect against Arb and checked the precision
trend; see hurwitz-diagnosis.txt and precision-diagnosis.txt in the build artifacts.
This corroboration is not a certified roundoff enclosure.
For truncation after K=120, let J=d-1, r=(a/q)^2 and
A=sum_j |c_j| q^j. Since zeta(p,q) <= q^(-p)(1+q/(p-1)) for real p>1, the
absolute remainder in the derivative is at most
A r^(K+1)/((K+1)(1-r)) [1+q/(2K+1-J)].
The check records this explicit truncation bound separately from its observed
finite-precision discrepancy. It tests finite nonzero controls first:
L_1=2 log(2 pi), L_2=1/6+4 log A (Glaisher–Kinkelin), and
L_3=log pi+zeta(3)/(2 pi^2). It also verifies the zeta functional equation
for every negative-even zeta derivative used in odd dimensions, and compares
all determinants with the rounded values of source Table 2.
The full run compares the generator's 150- and 200-digit ball evaluations,
checks finiteness and radius accuracy, and independently evaluates every
dimension. After upload, verify(sample=None) checks all stored entries, while
postfill.py checks their actual decimal intervals against the independent
continuation and the invertible convention relation.