bibkey: sanftenberg2026harmonicabel authors: David Sanftenberg year: 2026 title: “Signed harmonic Mobius tails against monotone finite weights” doi: null url: https://github.com/dbsanfte/RiemannGaussian/blob/24444671cee3bf643ff1307909b961a329372a9e/RiemannGaussian/MoebiusHarmonicMonotoneTail.lean claim: “The source identifies the literal harmonic Mobius prefix and retains both finite Abel endpoints; the local consumer uses existing Mathlib algebra to transport one-sided prefix-difference and exact signed-anchor budgets.” strata_touched:
- D5/S3/Arith/Robin/PrimePrefixMobiusDirectedAbel license: Apache-2.0 triage: anchor
Actual harmonic Mobius tails and source attribution
Immutable repository dbsanfte/RiemannGaussian revision
24444671cee3bf643ff1307909b961a329372a9e (6 October 2026).
Source RiemannGaussian/MoebiusHarmonicMonotoneTail.lean, SHA256
2477f924e258d30f80757227cf95c916eccdc1ed183bbe78150cfbb45e05a834,
Git blob c4c7067dc51126952722a9929552226f3dad2d2c.
The full source was read and its Git blob matched to the complete,
untruncated immutable tree. The root LICENSE says
Copyright 2026 David Sanftenberg; the selected source leaf itself has
no additional author/copyright header. The commit author and existing
Fermi-source attribution independently identify David Sanftenberg.
Exact source contracts and local consumption
sum_moebiusHarmonic_weight_eq_by_parts accepts every natural D<M and
arbitrary real b. With the actual H(N)=sum over 1<=k<=N of mu(k)/k,
it proves the finite endpoint identity
S(D,M,b)=b(M)H(M)-b(D+1)H(D)
-sum over D<n<=M-1 of (b(n+1)-b(n))H(n).
abs_sum_moebiusHarmonic_antitone_le requires b nonnegative and antitone
on [D+1,M] and a bound abs(H(n))<=e at every n in [D,M]. It proves
abs(S)<=2eb(D+1). The actual-prefix bound is an input of that theorem.
The local source directly calls adopted Mathlib’s Finset.sum_Ioc_by_parts
and Finset.sum_Ico_sub; it does not port or rename these generic suppliers.
The private harmonic range conversion and telescoping bodies are adapted
from the selected Gaussian source, with names/namespace changed. They are
consumed on the live path to the complete anchored identity and lower bounds.
The local consumer preserves Q_D(n)=H(n)-H(D) and the terminal term:
S=b(M)Q_D(M)+sum over D<n<=M-1 of (b(n)-b(n+1))Q_D(n).
Nonnegative antitone weights make every displayed coefficient nonnegative with total b(D+1). If every actual Q_D on D<n<=M is at least -e, e>=0, the new same-object finite comparison yields S>=-e*b(D+1). An absolute prefix bound on the full [D,M], including D, supplies the more precise nonnegative budget delta+H(D), and S>=-(delta+H(D))*b(D+1). The first-weight-zero case forces the whole finite tail to be zero; M=D+1 is already retained by the full identity with an empty inner sum. These are classical finite comparisons used on the literal Mobius object, without a claim of new mathematical priority for Abel summation.
The source is pinned to Lean4.33.1 / Mathlib
0df444a360eaa60ab8c11dca51a86af692955474; the local consumer uses
Lean4.33.0 / Mathlib db584cd6d46c92f209a44c0f1c829460d327499d.
No Gaussian harmonic-decay body is copied here. Original Robin kernel
matching, nonnegative antitone window or signed variation, odd/full
transport, critical normalization and infinite-tail acceptance remain open.
Licence and inspected NOTICE scope
The complete root LICENSE is 11,348 bytes, SHA256
b9e4bb798f04c0f434a665c1dcd27c761659b1dc07bab9b726252525a5d8260e,
Git blob 35abd0dd00c4df7825b73e2af6018788afab0d17, matched exactly to
this immutable tree. Its complete original copyright line is retained below.
The complete tree contains NOTICE files under HermitianRankTrace,
MontgomeryVaughan and vendor/zeta23; none is in the root or the selected
leaf’s immediate RiemannGaussian directory. Those unrelated supplies and
notices are not copied or imported by this local finite-algebra unit.
This states the inspected source scope and does not falsely assert that
the entire upstream repository has no NOTICE.
Complete upstream licence
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