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3504 bytes, as of the version from 2026-10-31 20:22 (current). Recorded here, not run.
# Source and convention checks for T639
Primary sources read in this build:
1. Bettin–Conrey, *Period functions and cotangent sums*,
https://arxiv.org/pdf/1111.0931v2, p. 3 and §2, pp. 11–13.
The definition has a minus sign and the Vasyunin conversion uses the
modular inverse of the numerator. Equation (12) supplies
D(0,h/k)=1/4+i c0(h/k)/2. Equation (13) and its following discussion
identify the finite sums in integer-dilate approximation Gram matrices.
Journal metadata and DOI were checked at https://arxiv.org/abs/1111.0931.
2. Báez-Duarte–Balazard–Landreau–Saias,
https://arxiv.org/pdf/math/0306251, §4: Vassiounine spelling and the
fractional-part context. Both papers are archived locally, with hashes
in source-manifest.json.
3. https://dlmf.nist.gov/5.5.E4, read through the web tool: the digamma
reflection formula. Direct download of the TeX variant was unavailable.
4. https://dlmf.nist.gov/25.11.E13 and §25.11 equation (14), read through
the web tool: Hurwitz zeta at zero and at nonpositive integers.
Direct download of the TeX variant was unavailable. These download
failures do not indicate that the mathematical source was unread.
## Independent Estermann computation
For Re(s)>1, expand d(n) by n=ab. Group a and b into their positive residue
representatives r,t=1,...,k. The resulting continuation is
D(s,h/k)=k^(-2s) sum_{r,t=1}^k exp(2 pi i h r t/k) zeta(s,r/k) zeta(s,t/k).
At s=0, use zeta(0,x)=1/2-x. Every coefficient is rational and every phase
is an exact root of unity. This gives an exact computation independent of
cotangent evaluation. In checks.py the integer coefficients are accumulated
first, then divided in the rational/cyclotomic field by 4k^2. Exact
conjugation yields c0=(D-conjugate(D))/i, and D+conjugate(D)=1/2 is checked.
No cotangent or generator sum is used to construct this independent answer.
## Family Hilbert-space conventions
This table stores only c0. To pin down the application selecting its modulus
bound, use the family's L^2((0,infinity),dx), rho_n(x)={1/(nx)} and
chi=1_(0,1). Define A(lambda)=integral_0^infinity {t}{lambda t}dt/t^2.
Substituting t=1/(jx) in the Gram integral gives
G_jk=A(j/k)/j. This substitution has dx=-dt/(j t^2).
Similarly b_n=(1/n) integral_(1/n)^infinity {t}dt/t^2. The contribution
from 1/n to 1 is log(n). On each [m,m+1], m>=1, the integral is
log((m+1)/m)-1/(m+1). Its sum for m=1,...,M is
log(M+1)-H_(M+1)+1, which tends to 1-gamma. Thus
b_n=(log(n)+1-gamma)/n. Endpoints have measure zero. These are the shared
normalizations, with natural logarithms, without anchoring or unit-norm
rescaling. No Gram matrix, minimizer, or additional error table was built.
## Live corpus checks
preflight.json records already_here/already_asked and the live family
neighbours; lookup.json is the successful API lookup evidence. The earlier
q= lookup attempts in preflight.json were refused syntax probes, not empty
results. Searches for Bettin, Vasyunin, Vassiounine and Nyman returned no
matching family. Sample c0(1/3), c0(1/4), c0(1/6) values match unrelated
existing quantities and do not duplicate this family. Tags number theory,
algebraic and special values exist. The proposed Nyman-Beurling criterion
tag is absent and was not created for a single table.
The coordinated skill manifest is
agents/runs/20261031T200151Z-build-skills/manifest.json.
Both archived versions have SHA-256
a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278.