A kernel-checked proof of Dusart's bound for the sum of 1/p from a published explicit Chebyshev estimate

Dusart′s bound Σ_(p≤x) 1/p-log log x-B≤1/(10 log²x)+4/(15 log³x) for x≥10372, where B is Mertens′ constant, rests on an explicit Chebyshev estimate |θ(t)-t|≤ t/(5 log²t) for t≥3 594 641, partial summation, and a computer check of the range below 3 594 641. We prove the bound in Lean 4 with Mathlib, with that Chebyshev estimate as the only hypothesis: the kernel checks the implication, and the proof uses no axioms beyond the standard three. This includes the identification of B=γ+Σ_p (log(1-1/p)+1/p) as the limit of Σ_(p≤x) 1/p-log log x, which we derive from the Euler product, the pole of ζ at 1 and Γ′(1)=-γ. It also includes the finite range [10372,3 594 641], covered by a certificate that the kernel evaluates: 821 632 integers coprime to 210, each kept as a possible prime or rejected by an explicit factor, fixed-point logarithms with proved error bounds, a certified lower bound B≥0.261497002, and 830 mesh cells. The certificate needs no primality proofs and uses no native_decide; its data are produced by a program and accepted only through kernel-checked evaluations of a checker whose soundness is proved. In this version the Chebyshev estimate itself is proved for 3 594 641≤ t≤3 584 912 847 by a second certificate, a segmented sieve of the odd numbers evaluated by the kernel with proved soundness (3 416 sieve segments, 58 309 mesh windows, an enclosure of θ(3 594 640) from log 2), so the bound for Σ_(p≤x) 1/p rests only on the estimate for t≥3 584 912 846=⌊e^22⌋. That remaining input is a single published, refereed table row: Broadbent, Kadiri, Lumley, Ng and Wilk prove |θ(t)-t|<0.15952 t/log²t for t>5·10^6, and the kernel checks the bound for Σ_(p≤x) 1/p with exactly this row as hypothesis. That row rests on rigorous verifications of the Riemann hypothesis to large height. We also report an interval computation showing that the zero-free half-plane Re s>7/8, now a formally proved theorem, together with the explicit formula of Cully-Hugill and Johnston, gives the needed estimate for t≥10^53.35; for ⌊e^22⌋≤ t<10^53.35 our argument continues to use the published row. A companion release proves the needed estimate in Lean for t≥2^626 from the formal half-plane alone. Both of Dusart′s thresholds are least integer thresholds for their inequalities: a separate computation in exact rational arithmetic with proved logarithm bounds shows, from the certificate′s kernel-checked enclosure of θ(3 594 640), that the Chebyshev estimate fails at t=3 594 640, and, with an upper bound for B, that the bound for Σ_(p≤x) 1/p fails at x=10371.

Companion to doi:10.5281/zenodo.23400116 and doi:10.5281/zenodo.23424383.

Contents of the release

• paper/ — the manuscript (LaTeX source and PDF) and dusart_formal_gen.py, which reads every stated count and constant from the Lean project and the interval log and writes dusart_formal_numbers.tex.
• lean/ — the Lean 4 project (Mathlib pinned in lakefile.lean): the analytic part, the certificate checker with its soundness proofs, the 24 certificate data modules (DusartLean/Data/), the assembly DusartLean/Final.lean; the kernel sieve with its soundness proofs (DusartLean/Theta/), the 221 data and check modules of the Chebyshev certificate (DusartLean/Theta/Gen/) and its assembly DusartLean/Theta/Final.lean; and Audit.lean, which prints the statement and axioms of the main theorems.
• programs/dusart_gen/ — the Rust generator of the certificate data (GMP integers via rug); it reproduces the shipped data modules and assembly byte for byte. The data are checked by the Lean kernel, not trusted.
• programs/theta_cert_gen/ — the Rust generator of the Chebyshev certificate (segmented sieve, fixed-point logarithms, greedy windows); it reproduces the shipped modules in DusartLean/Theta/Gen/ byte for byte.
• programs/dusart_finite/ — the MPFR interval program that measures the margin of the bound on the finite range; its output is paper/dusart_formal_runs/dusart_finite.txt. Not part of the proof.
• REPLAY.md and replay/ — a clean replay from this bundle alone: 10 of 10 steps pass.
• SHA256SUMS — the SHA-256 of every file.

Verification

In lean/ (Lean v4.33.1 via elan): lake exe cache get, then build the modules listed in DusartLean.lean one at a time (each certificate module needs at most about 3.5 GB of memory; build the Chebyshev certificate modules strictly one at a time), then lake env lean Audit.lean: every listed theorem reports only propext, Classical.choice, Quot.sound. In programs/dusart_gen/: cargo run --release -- <dir> 4000 36000 regenerates the data modules, and in programs/theta_cert_gen/: cargo run --release -- <dir>. In programs/dusart_finite/: cargo run --release -- 3594641 100000000. In paper/: python3 dusart_formal_gen.py --check. Integrity: shasum -a 256 -c SHA256SUMS.

Public verification record

The development is also published on the Prove2Me formalization platform, where every proof was re-checked by the platform's Lean server: the main theorem https://prove2.me/theorems/80f743a0-d076-44fe-97e7-d8c1c4ae5f43 (Proved), the finite range https://prove2.me/theorems/afb22cde-adb1-4307-9ed9-db8634f3b1ef (Proved), the Chebyshev estimate as an open problem https://prove2.me/theorems/e4af78fd-cb92-428c-ac7b-1afa1eeb4e3e, and the unconditional bound, reduced to that estimate, https://prove2.me/theorems/7a9cd00f-d5f7-4cea-b215-e0977a43a746. Of the Chebyshev certificate, the checker definition and the base range https://prove2.me/theorems/75c2d07d-adb9-4c86-9766-8a3fe000870e are proved there; the finite part of the estimate https://prove2.me/theorems/302a7138-f59c-4188-b234-81b54a5bf65f is proved in this release (its mesh exceeds the platform's per-job limits), and the large part https://prove2.me/theorems/6beffd1a-8a12-47cc-a6af-b7880427fe93 is open.

Hardware

One MacBook Pro (Apple M3 Pro, 11 CPU cores, 18 GB memory); no GPU, no cluster.

License: Licensed by SmartLedger Solutions under CC BY-NC 4.0 (LICENSE.md). Commercial use requires a written license from SmartLedger Solutions (labs@smartledger.solutions). No patent license is granted.