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Cross-Energy of Dirichlet Partial Sums. Part 18: The Euler-Caceres bound

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

Contributed by Pedro Caceres

Version 2 / Oct 18, 2026 / CC BY 4.0

Abstract

Gonek, Hughes and Keating factored ζ≈PXZX\zeta\approx P_XZ_X, an Euler product times a product over the zeros. We use an exact form ζ=EX∗ZX\zeta=E^*_XZ_X, EX∗E^*_X a smoothed Euler factor, and for $\re s_0>\frac12$ study the Euler--Caceres bound BX(s0)=12log⁡∫∣ZX(12+it)∣2Ps0(t) dt−log⁡∣ZX(s0)∣B_X(s_0)=\frac12\log\int|Z_X(\frac12+it)|^2P_{s_0}(t)\,dt-\log|Z_X(s_0)|, Ps0P_{s_0} the Poisson kernel. It is the off-line Blaschke term of the Poisson--Jensen formula at s0s_0, independent of XX, plus a positive Jensen gap. Under RH, (log⁡X)BX(s0)→∑γκm(γ)Ps0(γ)(\log X)B_X(s_0)\to\sum_\gamma\kappa_{m(\gamma)}P_{s_0}(\gamma), with universal constants κm\kappa_m for multiplicity mm (κ1=1.8279…\kappa_1=1.8279\ldots). Hence, for any single s0s_0, RH holds if and only if BX(s0)=O(1/log⁡X)B_X(s_0)=O(1/\log X), and RH with simple zeros if and only if (log⁡X)BX(s0)(\log X)B_X(s_0) tends to an explicit number (0.040186…0.040186\ldots at s0=2s_0=2). If $\Theta=\sup\re\rho>\frac12$, Landau's theorem in the variable log⁡X\log X gives BX(s0)=Ω+(XΘ−12−ε)B_X(s_0)=\Omega_+(X^{\Theta-\frac12-\varepsilon}) for $\re s_0>1$, attained or not, so RH holds iff BX(s0)≪XεB_X(s_0)\ll X^\varepsilon at one point. Unconditionally, BX(s0)=O(X/log⁡X)B_X(s_0)=O(\sqrt X/\log X) for $\re s_0>1$, and the off-line term at s0=2s_0=2 is below 2.3⋅10−122.3\cdot10^{-12}. Under RH we identify the second-order term up to an interaction of the wells (known only to be o(log⁡X)o(\log X)), related heuristically to Montgomery's pair correlation; it is governed by the gaps between zeros, and under a small-gap hypothesis the law holds with error O(log⁡log⁡X/log⁡X)O(\log\log X/\log X). We give a function-field model where the bound holds, and controls (a Beurling system with the zeros of ζ\zeta and a zero at 0.60.6, the Davenport--Heilbronn function) suggesting that a proof must couple the Euler product with the functional equation. Numerically the bound follows the law, stays below the floor 0.03860.0386 of Dirichlet-polynomial mollifiers at the centre s0=1s_0=1, and grows for the Davenport--Heilbronn function and for a planted off-line zero. We do not prove RH.

Author comments: Supersedes hexagon:2610.00343 (An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis), whose theorems are here in the same or a stronger form. Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). Version 2. 52 pages, 8 figures, 18 tables; ancillary Python scripts reproduce the tables and figures. Part 18 of the book-series Cross-Energy of Dirichlet Partial Sums (thirty-eight parts, Parts 0-46; block IV: Weil positivity; Part 0 is the guide); formerly Part 11 (renumbered 2 Nov 2026). We do not prove the Riemann hypothesis.

Provenance statement

The research program is the author's: the X-Y decomposition and cross-energy C2 of the Dirichlet partial sums (consolidated paper, doi:10.5281/zenodo.23243641), the smoothed factorization of zeta into a prime factor and a zero factor (the "Euler-Caceres" factor), and the aim of bounding zeta on the critical line in a way that distinguishes it from every other line. The author used Claude (Anthropic), an AI assistant, as a research and writing aid. Claude derived and checked the proofs, ran the numerical computations in Python, and drafted the LaTeX text and figures. A separate AI review agent with no part in the drafting refereed the proofs; all reported issues were fixed, including a restatement of the main corollary. The author directed the work, checked the results and takes full responsibility for the content. No AI system is an author. The paper does not claim a proof of the Riemann hypothesis.

Tools used

Anthropic
Claude OpusVersion 5.5
Python
PythonVersion 3.13

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Version history

  1. v1 · 11 pagesSubmitted by Pedro CaceresInitial depositSupersededOct 18, 2026
  2. v2 · 52 pagesSubmitted by Pedro CaceresSupersedes hexagon:2610.00343 (An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis), whose theorems are here in the same or a stronger form. Revised 10 Nov 2026 after external examination (see CHANGES_v2.md). Version 2. 52 pages, 8 figures, 18 tables; ancillary Python scripts reproduce the tables and figures. Part 18 of the book-series Cross-Energy of Dirichlet Partial Sums (thirty-eight parts, Parts 0-46; block IV: Weil positivity; Part 0 is the guide); formerly Part 11 (renumbered 2 Nov 2026). We do not prove the Riemann hypothesis.CurrentNov 11, 2026