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Cross-Energy of Dirichlet Partial Sums. Part 33: Optimised mollifiers and the bound Λ≤0.174\Lambda\le0.174 for the de Bruijn--Newman constant

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.CV — Complex Variables

Contributed by Pedro Caceres ↗

Version 1 / Nov 13, 2026 / CC BY 4.0

Abstract

The Polymath project reduced upper bounds for the de Bruijn--Newman constant Λ\Lambda to three verifiable hypotheses: the Riemann hypothesis up to height X/2X/2, a zero-free region for Ht0H_{t_0} to the right of XX, and a barrier at XX. With the verification of the Riemann hypothesis up to 3⋅10123\cdot10^{12} by Platt and Trudgian, this gave Λ≤0.2\Lambda\le0.2. The bottleneck is the zero-free region, which Polymath established with Euler-product mollifiers. We replace the Euler product by a mollifier ∑d≤Dλdd−s∗\sum_{d\le D}\lambda_d d^{-s_*} whose weights are chosen by solving an ℓ1\ell^1-regression problem, so that any choice of weights still yields a valid lower bound. At the same height X≈6⋅1012X\approx6\cdot10^{12} and t0=0.155t_0=0.155, y0=0.2y_0=0.2, the lower bound for the mollified sum improves from −0.12-0.12 (our Lemma~\ref{lem:moll} evaluated with the Euler mollifier on the primes up to 1313) to +0.05+0.05 with D=120D=120; with D=240D=240 it is still +0.04+0.04 at t0=0.154t_0=0.154. We cover all x≥Xx\ge X by a short chunking argument together with a tail estimate, verify the barrier with Polymath's Taylor scheme, and obtain \[ \Lambda\le 0.174 . \] The certificates are computer-assisted. They use Arb ball arithmetic for all transcendental functions and outward-rounded floating-point arithmetic for the large sums. The code, the weights and the logs are included, and the certificate was reproduced on a second platform. We also report that at this height the weights we found with D=240D=240 do not reach 0.1730.173, and we compare with Polymath's table of conditional results. We do not prove the Riemann hypothesis.

Author comments: Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). 20 pages, 3 figures. The ancillary scripts are those of the original part (ancillary zip in the part folder). Part 33 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly Part 13. Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.

Provenance statement

The research programme is the author's. The paper was written in November 2026 and is Part 33 of the book-series Cross-Energy of Dirichlet Partial Sums (Part 13 in an earlier numbering). The theorem is computer-assisted: it rests on Polymath 15's criterion, on the verification of RH to 3*10^12 by Platt and Trudgian, and on an interval-arithmetic certificate (Arb via python-flint) whose scripts and logs are in the ancillary files. Claude (Anthropic) was used throughout as a research and writing aid: to derive and check proofs, to write and run the computations, to search the literature, and to draft text and figures. On 10 November 2026 the paper was examined twice by independent AI referee sessions that had not seen it being written; each report listed errors and recomputed the numbers it could, and this version contains the corrections from both rounds. It has not been refereed by a human reviewer. The author checked the results and is responsible for the content. The certificate was not re-run after the revision of 10 November 2026, which changed the exposition and two proofs (the covering lemma and the zero-counting proposition) but not the certified quantities. Bounds announced by other authors in 2026 are quoted as unrefereed claims and are not used in the proof.

References

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  9. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 18: The Euler--Caceres bound, preprint, 2026.
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  11. P. Caceres, Cross-Energy of Dirichlet Partial Sums. Part 35: How long can an off-line zero survive the de Bruijn--Newman heat flow? An explicit lower bound for Λ\Lambda forced by a zero off the critical line, preprint, 2026; an earlier version is hexagon:2610.00207.hexagon
  12. P. Caceres, Proof material for ``Optimised mollifiers and the bound Λ≤0.174\Lambda\le0.174 for the de Bruijn--Newman constant'' (version 3): mollifier weights, certificate scripts and complete certificate logs , Zenodo, 2026, .

Version history

  1. v1 · 20 pagesSubmitted by Pedro CaceresRevised 10 Nov 2026 after external examination (see CHANGES_v2.md). 20 pages, 3 figures. The ancillary scripts are those of the original part (ancillary zip in the part folder). Part 33 of the book-series Cross-Energy of Dirichlet Partial Sums (47 parts, Parts 0-46); formerly Part 13. Renumbered 2 Nov 2026; mathematics unchanged. The paper states once, in its discussion, that it does not prove the Riemann hypothesis.CurrentNov 12, 2026