bibkey: pntplus2026mertens authors: Harold G. Diamond and Janos Pintz year: 2009 title: Oscillation of Mertens’ product formula doi: 10.5802/jtnb.687 claim: Mertens’ third theorem (1874) is stated by Diamond and Pintz (2009), equation (1.1); the pinned PrimeNumberTheoremAnd Lean source formalizes this asymptotic with the standard axiom closure. strata_touched:

• D5/S3/Weil/Mertens/Estimates
• D5/S3/Weil/Mertens/LogZeta
• D5/S3/Weil/Mertens/Gamma
• D5/S3/Weil/Mertens/Third license: citation-only triage: anchor

Oscillation of Mertens’ product formula

Harold G. Diamond and Janos Pintz, Journal de Theorie des Nombres de Bordeaux 21 (2009), no. 3, pp. 523-533. Mertens (1874); literature attestation via Diamond and Pintz (2009), equation (1.1), p. 523: (product over primes p <= x of (1 - 1/p)) * log(x) -> exp(-gamma). This is the classical asymptotic formalized by the pinned upstream Lean port recorded below. The citation attests that formula; the port’s exact error bounds and Lean axiom closures are supported by the recorded code measurements, not by this paper’s separate oscillation theorem.

DOI verification and route

On 2026-09-07, https://doi.org/10.5802/jtnb.687 resolved with HTTP 200 to https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.687/. The publisher’s metadata confirms the authors, title, publication year 2009, volume 21, issue 3, pages 523-533 and DOI. Its linked PDF was opened and equation (1.1) checked on printed page 523. The online publication date is 2010-03-22; the journal volume and article are dated 2009.

Route A retains this single note at the address cited by the four frozen modules’ NOTICE comments. Their anchors: [] fields and comments cannot be edited in this repair. Reusing apostol1976introduction by deleting this note, or moving the NOTICE out of L, would require changing those frozen links. The existing Apostol DOI was independently resolved to its Springer book page; it is not duplicated here. This note instead cites the distinct paper above. The citation-only license metadata concerns that paper. The upstream port’s Apache-2.0 notice and complete license remain below with its measurements.

Mertens III Compatibility Measurement

Provenance: codex-cli implementation seat mertens3-0907/attempt-1, dispatched by the consensus-rnd runner. No additional skill or review seat was used by this worker; all measurements below are this worker’s direct readings. This is a literature port for the user-selected third-tier Robin/Gronwall research line, not a new mathematical result.

Source and Route

• Repository: https://github.com/kimihiro64/PrimeNumberTheoremAnd.
• Immutable commit: 6a380f0c4658c04a420a9eb00b1ed62a1e3fde01.
• File: PrimeNumberTheoremAnd/IEANTN/Mertens.lean.
• The downloaded bytes equal the previous probe’s UpstreamMertens.lean (cmp, exit 0). That probe recorded the AlexKontorovich repository name; both citations identify the same commit and source bytes in this reading.
• Reused mertens-probe-0907/attempt-1/MertensCompat.lean, its runmake.mjs, and its additive probe.mk. The normal root lean recipe is retained.
• Route PrimeNumberTheoremAnd.EulerMaclaurin to the existing D5.S3.Weil.ZetaPntBase.EulerMaclaurin; remove Architect blueprint metadata and one redundant rfl, as already done by the previous probe.

Q1 Reading

Repository HEAD at measurement: c32b86362c4bf8d27c07ffbfe23bca128f90b070. Lean v4.33.0; Mathlib db584cd6d46c92f209a44c0f1c829460d327499d.

Command: make -f Makefile -f <attempt>/probe.mk lean PROBE=<attempt>/Q1.lean. Exit: 0. Wall time: 160.490721292 seconds, including the root project build. The initial cache receipt reported both Mathlib and project layers warm; this is not a cold-build timing or a prediction for CI.

The three task signatures were independently written as example types and closed by the corresponding upstream constants, with no additional hypotheses. All three type checks passed. The exact axiom output is:

'Mertens.E₃.abs_le' depends on axioms: [propext, Classical.choice, Quot.sound]
'Mertens.E₃.bound''' depends on axioms: [propext, Classical.choice, Quot.sound]
'Mertens.E₃.bound'''' depends on axioms: [propext, Classical.choice, Quot.sound]

The final quote in each line closes Lean’s quoted declaration name. There is no sorryAx or custom axiom in any of these three closures. One deprecation warning names Set.mem_setOf_eq; it is not an analytic gap.

Artifacts: /var/folders/7r/h8yjr2y927n8m2kh38c18n9w0000gp/T/consensus-rnd/sshx/mertens3-0907/attempt-1/. The exact source, signatures and build output are Q1.lean, q1.make.log, q1.source.lean and q1.receipt.json in that directory.

Repository Search

At the measured HEAD, git grep -n -P '\b(Mertens|Gronwall|ChebyshevMertens)\b' -- D5 found weak product estimates in PrimeGaps/EulerProducts, hypotheses in Weil/ZetaCore/Hypotheses, and provenance in the Euler-Maclaurin port. These are not the sharp product asymptotic. The same word-boundary feature was checked by git grep -l -P '\btheorem\b' -- 'D5/**/*.lean': 3613 files. These are lexical search readings, not a proof of global nonexistence.

Q1 establishes compatibility of the external source. It does not by itself freeze a repository theorem or establish the Gronwall upper envelope.

Port Selection and Scope

Form A: source port. Lean.parseImports' recursively measured the routed non-Mathlib closure: 2 files, 2557 lines. Of these, the 102-line Euler-Maclaurin module is already frozen in D5; the external source is 1 file, 2455 lines. Architect is metadata only and was removed by the previous successful port. The dependency traversal stops at the pinned Mathlib boundary.

The three endpoints’ elaborated constant dependency graph was then traversed through both declaration types and proof values. findDeclarationRanges? located 111 source declarations covering 1731 lines. Only these declarations were selected, preserving proof bodies and source order. The result is four modules: Estimates, LogZeta, Gamma and Third. The five private helpers used across those boundaries are now public. One deprecated Set lemma name was updated; the prior probe’s redundant-rfl correction is retained.

This is a small, buildable import closure. Splitting at the analytic interfaces keeps each module below the repository’s 800-line hard limit. It introduces no package dependency and no new analytic hypothesis. Import closure and declaration-selection receipts are closure.make.log, declaration-closure.json and port-transform.json in the attempt directory.

Final parsed closure after declaration pruning: 4 new files, 2014 lines (Estimates 587, LogZeta 360, Gamma 699, Third 368). The Euler-Maclaurin import is no longer required by this smaller declaration set and was removed. The original routed whole-source Q1 measurement remains unchanged. The four formal-unit modules plus this source/license note total 5 paths. Canonical freezing adds four event files and four state files: the complete change is 13 paths, equal to the user-supplied PR p75. These four modules form one endpoint’s proof closure; no unrelated theorem family is included.

make lean on the split port: exit 0, 35.4733835 seconds. After removing the unused import: exit 0, 78.079721041 seconds. PortCheck.lean then independently imported Third, rechecked all three exact task signatures and printed all three standard axiom closures: make exit 0, 14.9107345 seconds. Its import parser produced the final counts above. These are warm local timings including the root build; upstream formatting warnings remain nonblocking.

Canonical make lean-report exited 0 in 335.633129708 seconds, with both cache layers warm and mode=full-fallback. The report SHA-256 is c174d6f10e7cfeedd2ee67b1c85302185fc6035ca2f6766acee16833d28ebf99. ledger-align then exited 0 with selectors_considered=5 changed=0 added=4 unchanged=1 conflicts=0. The existing Euler-Maclaurin module was the unchanged selector used to limit the command’s scope; it is not an import of the final port. All eight generated ledger paths belong to the four Mertens modules. No Gronwall atom is covered.

Before opening the PR, origin/dev at 03b70412c96c6c35768a194dfe1f865b93c59596 was searched again using the same word-boundary query. It added only a Robin padding comment to the earlier nearby hits; no sharp Mertens III declaration was found in that searched scope. The matching-feature positive control returned 3617 D5 Lean files. git merge-tree --write-tree origin/dev HEAD exited 0; the changed-path intersection with paths deleted on dev since the initial base was empty.

proof_shape: bind-only; admission_basis: rule-11-upstream-wrapper. The named upstream declarations are Mertens.E₃.abs_le, Mertens.E₃.bound'' and Mertens.E₃.bound'''. The concrete API requirement is the sharp prime-product estimate needed by the Gronwall upper-envelope consumer on the user-selected Robin line. Every retained helper is in the elaborated dependency closure of at least one of those endpoints, in the consumer-to-prerequisite direction. computational_content.kind: none: these are general analytic estimates, not finite certificates or numerical reductions. This port does not establish Robin’s inequality or Gronwall.

Gronwall Upper-Envelope Gap

Let E(y) = product (p prime, p <= y), (1 - 1/p)^(-1) and R(n) = sigma(n)/(exp(gamma) * n * log(log(n))). The following is a consumer-to-prerequisite plan, not a claim that the unmeasured steps compile. self means a repository proof is still needed, not a novel mathematical result.

No PNT or RH premise enters this upper-envelope route. Mertens removes the sharp-constant analytic dependency; it does not supply SigmaSplit. The equality limsup sigma(n)/(n*log(log(n))) = exp(gamma) additionally needs the lower limsup construction, for example integers with sufficiently saturated small prime powers. That lower half is separate and was not proved in this attempt.

Measured leaf, in GronwallLeaf.lean (external probe, not an independent deposit):

theorem sigma_prime_pow_ratio_le {p : Nat} (hp : p.Prime) (a : Nat) :
    (ArithmeticFunction.sigma 1 (p ^ a) : Real) / (p : Real) ^ a <=
      (1 - 1 / (p : Real))⁻¹ := by
  have hp1 : (1 : Real) < p := by exact_mod_cast hp.one_lt
  have hp0 : (0 : Real) < p := lt_trans zero_lt_one hp1
  have hpow : (0 : Real) < (p : Real) ^ a := pow_pos hp0 a
  have hfactor : (ArithmeticFunction.sigma 1 (p ^ a) : Real) =
      ∑ i ∈ Finset.range (a + 1), (p : Real) ^ i := by
    exact_mod_cast ArithmeticFunction.sigma_one_apply_prime_pow hp (i := a)
  rw [hfactor, geom_sum_eq (ne_of_gt hp1)]
  calc
    ((p : Real) ^ (a + 1) - 1) / (p - 1) / p ^ a <=
        (p : Real) ^ (a + 1) / (p - 1) / p ^ a := by
      gcongr
      linarith
    _ = (1 - 1 / (p : Real))⁻¹ := by
      rw [pow_succ]
      field_simp
      <;> ring

Command: make -f Makefile -f <attempt>/probe.mk lean PROBE=<attempt>/GronwallLeaf.lean. Exit: 0. Wall time: 56.590490625 seconds, warm local tree, root build included. Stuck at: none in this leaf; SigmaSplit remains unmeasured. The optional final ring generated an unused-tactic warning because field_simp already closed the goal; the measured source is retained unchanged.

'GronwallLeaf.sigma_prime_pow_ratio_le' depends on axioms: [propext, Classical.choice, Quot.sound]

Search scope: D5 sigma/prime-factor/product candidates, pinned ArithmeticFunction/Misc.lean, Algebra/Field/GeomSum.lean, Algebra/Order/Field/GeomSum.lean, and the inspected upstream Mertens source. The exact sigma bound was not found in those searched candidates. The proof reuses the two exact identities found in Mathlib, rather than re-proving them.

Retirement condition: once this repository’s own pinned Mathlib contains equivalent declarations, replace the matching source port with imports and applications of those declarations. Upstream acceptance alone is not the trigger.

NOTICE

This distribution contains portions of PrimeNumberTheoremAnd, copyright its contributors, licensed under the Apache License, Version 2.0. Source commit: 6a380f0c4658c04a420a9eb00b1ed62a1e3fde01; source file: PrimeNumberTheoremAnd/IEANTN/Mertens.lean. The upstream tree contains LICENSE and no separate NOTICE file (recursive GitHub tree, truncated=false, inspected by its structured entries).

Modified by trureturing on 2026-09-07: repository import routing, removal of Architect metadata, restriction to the E3 proof dependency closure, module splitting, visibility of five cross-module helpers, removal of one redundant rfl, and replacement of Set.mem_setOf_eq by Set.mem_ofPred_eq. The preliminary routing and tactic correction reuse the earlier mertens-probe-0907/attempt-1 worker’s compatibility source.

The upstream exposition attributes the mathematical arguments to Leo Goldmakher, A quick proof of Mertens’ theorem: https://web.williams.edu/Mathematics/lg5/mertens.pdf. It also cites Arend Mellendijk’s earlier unfinished formalization: https://github.com/FLDutchmann/Analytic/blob/main/Analytic/Mertens.lean. These attributions do not claim that either author reviewed this port.

The complete upstream license follows verbatim.

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