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An Euler-product mollifier for the Jensen moment of zeta and an energy criterion for the Riemann hypothesis

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

Contributed by Pedro Caceres ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

The Jensen moment of Balazard, Saias and Yor expresses the zeros of the Riemann zeta function outside the critical line through the mean value of the logarithm of its absolute value on that line. Jensen's inequality converts any mollifier into an upper bound for this contribution. For Dirichlet-polynomial mollifiers, the bound cannot fall below 0.0386, a barrier arising from the pole at s = 1. We instead use the reciprocal of a smoothed Euler factor, denoted E-star(X,s), constructed from prime powers up to a constant multiple of X together with an explicit pole term. The quotient of the zeta function by this factor is then expressed solely in terms of its nontrivial zeros. We prove that the resulting bound B(X,s0), for any center s0 with real part greater than one half, tends to zero under the Riemann hypothesis (RH). Consequently, RH is equivalent to the convergence of B(X,s0) to zero. If the supremum of the real parts of the zeros is attained and exceeds one half, the bound grows at least as fast as X to a positive power divided by log X. Under the same attainment assumption, RH is therefore equivalent to boundedness, as X tends to infinity, of a single weighted mean-square energy: the integral over the real line of the squared absolute value of zeta divided by E-star on the critical line, weighted by 1/(1+t squared). Numerically, the bound decreases to 0.0015 at X = 10,000 for the Balazard–Saias–Yor center s0 = 1. At s0 = 2, it decreases approximately as 0.037 divided by log X up to X = 1,000,000, close to a heuristic prediction involving the constant kappa = 1.828... . By contrast, the corresponding quantity grows for the Davenport–Heilbronn function. We describe the geometry of the factorization and use two control functions to identify the additional ingredients required for a proof of boundedness of the mean-square energy.

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

References

  1. H. M. Bui, S. M. Gonek, M. B. Milinovich, A hybrid Euler--Hadamard product and moments of ζ′(ρ)\zeta'(\rho), preprint, arXiv:1302.5032, 2013.arXiv
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  3. P. Caceres, The cross-energy of Dirichlet partial sums and the zeros of the Riemann zeta function, preprint, Zenodo, 2026, 10.5281/zenodo.23243641, Hexagon: hexagon:2610.00161; statement numbers refer to version 3 (2026).hexagonDOI
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  8. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function , 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, 1986.

Version history

  1. v1Submitted by Pedro CaceresInitial depositCurrentOct 18, 2026