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6334 bytes, as of the version from 2026-11-01 08:46 (current). Recorded here, not run.
# Numerical evidence for T646
The recorded constant is
S = 0.7825279853253842345766884742837840768058009796525991599926472087826073767199003522927165598264149835.
This is a 100-significant-digit decimal. NumberDB interprets it as an interval
of one unit in the last place on either side. It is not an exact rational.
No irrationality claim is made. No assumption about the Riemann hypothesis
is used.
## Definition and convergence
Write a_n = 1 - 1/n^2. The finite binomial definition of c_k and the absolutely
convergent reciprocal-zeta Dirichlet series give
c_k = sum(n>=1) mu(n) a_n^k / n^2,
with a_n^0 = 1, including n=1. For every nonnegative integer K, finite geometric
summation gives
sum(k=0..K) (-1)^k c_k
= sum(n>=1) mu(n) [1 - (-a_n)^(K+1)] / (2n^2-1).
The summands in absolute value are bounded by 2/n^2 independently of K.
Dominated convergence therefore proves convergence in the ordinary partial-sum
sense to
S = sum(n>=1) mu(n)/(2n^2-1).
This arithmetic series is absolutely convergent. Expanding its denominator
into sum(m>=1) 2^(-m)n^(-2m) is an absolutely convergent double series, and hence
S = sum(m>=1) 1/(2^m zeta(2m)).
This is Cisło and Wolf, arXiv:0807.2971, §4, equation (49), with the coefficient
convention of equations (4) and (14). These finite-sum and domination arguments
avoid assuming absolute convergence of the double series in k,n at t=-1.
## Main enclosure
`generate.py` sums m=1,...,400 using Arb real balls at 413 bits. Since
0 < 1/zeta(2m) < 1, its positive omitted tail is smaller than 2^(-400).
The code adds a symmetric ball of that radius, also containing the positive
tail. The result retains 398 relative accuracy bits. Its printed enclosure
has radius at most 6.34e-121. Both numerical rounding and truncation are covered.
The 130-digit repeat retains 503 relative accuracy bits.
## Independent calculation
`checks.py` starts from the arithmetic kernel mu(n)/(2n^2-1), forms the terms for n<=A=16
as exact rationals before ball conversion, and uses L=60 geometric terms only for n>A:
sum(n<=A) mu(n)/(2n^2-1)
+ sum(m=1..L) 2^(-m) [1/zeta(2m) - sum(n<=A) mu(n)/n^(2m)].
The omitted absolute tail is at most
1 / [2^(L+1) (1-1/(2(A+1)^2)) (2L+1) A^(2L+1)].
Indeed, for each n>A the remaining geometric series is
(2n^2)^(-L-1)/(1-1/(2n^2)); the denominator is bounded using n>=A+1,
and sum(n>A)n^(-2L-2) is bounded by its integral from A to infinity.
All bounds are exact rationals before outward ball conversion.
The independent calculation uses no Arb zeta or pi evaluator. It constructs
B_0,...,B_120 by the exact Bernoulli recurrence, with B_1=-1/2, and evaluates
1/zeta(2m)=2(2m)!/((2pi)^(2m)|B_(2m)|). This is DLMF 25.6.2. It encloses pi by
16 arctan(1/5)-4 arctan(1/239), using exact rational alternating-series
remainders after 162 and 48 terms. Arithmetic runs at 700 bits; the final
independent enclosure retains 550 relative accuracy bits. Its printed radius
is at most 5.24e-166. Thus it has ample precision to check the written interval,
not merely a matching midpoint.
The main, independent and higher-precision enclosures all lie strictly inside
the 100-digit last-place interval. Both sides are explicitly required to be
finite and narrow. All computation scripts and exact rational bounds are
attached; the independent method shares Arb arithmetic infrastructure but
neither the main series code nor its zeta evaluator.
## Controls and additional checks
- The exact Bernoulli recurrence returns B_0,...,B_4 and B_12=-691/2730.
- Exact Gaussian rational multiplication verifies Machin's angle identity.
Its pi enclosure agrees with Arb's pi, at over 680 relative accuracy bits.
- The Möbius sieve returns the known first ten values and satisfies
sum(d|n)mu(d)=0 for 2<=n<=10000, with value 1 for n=1.
- The independently evaluated reciprocal zeta values at 2, 4 and 6 agree
with 6/pi^2, 90/pi^4 and 945/pi^6.
- Adding ten units of the written last place is rejected by containment.
A radius-one ball is rejected as well.
- A direct, unaccelerated 10000-term Möbius sum agrees within its rigorous
absolute tail bound 1/10000. This coarse check is supplementary.
- Finite alternating coefficient sums through K=0,1,8,32,256 agree with the
finite arithmetic-kernel sums within the rigorous tail bound 2/10000.
Fresh coefficients are recomputed in 700-bit balls; none are treated as
exact downloaded decimals.
- The 87 decimal digits visible in OEIS A345412 agree. They are an external
comparison, not the source of the 100-digit error certification.
- The repository's dry_run.py passes exactness, precision-form, prose and
ring-import checks: one entry, longest raw ball string 138 characters,
about 0.2 KB. The stored decimal itself is 102 characters.
`checks-pre-fill.json`/`checks-evidence.json` records the local checks.
The separate `stored-checks-evidence.json`, attached after filling, records
verification on database readback and comparisons with all 257 live T644
coefficient intervals. `checks.py` repeats these with --stored. The final
handoff records the audit and attachment fingerprints. No independent critique
acceptance or human review is implied by these build checks.
## Sources and policy
Checked sources:
- https://arxiv.org/html/0807.2971, equations (4), (14), (47), (49), §4.
A downloaded HTML copy is retained in the run work directory.
- https://oeis.org/A345412, title, defining formulas, displayed digits.
The separate b-file was unavailable; only the visible 87 digits were used.
Its unrelated ordinary-sum comment and its integral formula are not adopted.
- https://dlmf.nist.gov/25.6, equations 25.6.1 and 25.6.2, read through the web
tool; the TeX endpoint did not return successfully.
- Live NumberDB documents T644, T645, T14 and T13, retrieved through the API.
Actual companion URLs were read from their authenticated attachment pages:
Báez-Duarte_coefficients and Values_of_the_Riesz_function.
Policy manifest: agents/runs/20261101T082247Z-build-skills/manifest.json.
Archived local and live skills were byte-compared and both have SHA-256
`a2b28d9953264d7b0ca62663785c2952190c5be6d6d3597fd695d9a54d606278`.
Environment: SageMath 10.9, numberdb 0.1.10. Every Sage computation in this run
used agents/sage.sh. No source, skill, published table, or deployment was edited.