A kernel-checked proof of Dusart's bound for the sum of 1/p from the 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(3 594 641-1) from log 2), so the bound for Σ_(p≤x) 1/p now rests only on the estimate for t≥3 584 912 846=⌊e^22⌋.

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.