Which Selberg axiom carries the Weil negativity? A Ramanujan-violating control

ζ(s+δ)ζ(s−δ) keeps an Euler product and a functional equation but breaks the Ramanujan bound, together with the pole and gamma-factor axioms; every zeta zero becomes an off-line pair at depth δ, and there are no composite terms. Its localised Weil functional is certified negative at the first ten zeta ordinates, carried by the prime-inflation channel, and equal to the pair's own term −4δ²eδ²/2d²: the pair depth is the inflation exponent. Over 600 zeta ordinates the sign follows a threshold δc(t) that rises with height. With the Davenport–Heilbronn and Epstein notes this gives a four-row table of which axiom's failure shows up in which arithmetic channel. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, the scripts and result files it cites, and the dated literature-search log behind its priority statements.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-10-30
DOI
https://doi.org/10.5281/zenodo.23812169
Primary Topic
Analytic Number Theory Research
Type
preprint
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Which Selberg axiom carries the Weil negativity? A Ramanujan-violating control

Nathan Berzai
7 links
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

Which Selberg axiom carries the Weil negativity? A Ramanujan-violating control

Nathan Berzai
preprint en

Abstract

ζ(s+δ)ζ(s−δ) keeps an Euler product and a functional equation but breaks the Ramanujan bound, together with the pole and gamma-factor axioms; every zeta zero becomes an off-line pair at depth δ, and there are no composite terms. Its localised Weil functional is certified negative at the first ten zeta ordinates, carried by the prime-inflation channel, and equal to the pair's own term −4δ²eδ²/2d²: the pair depth is the inflation exponent. Over 600 zeta ordinates the sign follows a threshold δc(t) that rises with height. With the Davenport–Heilbronn and Epstein notes this gives a four-row table of which axiom's failure shows up in which arithmetic channel. Written jointly with Claude, an AI system made by Anthropic, which carried out much of the computation, the derivations and the drafting; the human author is responsible for the content. Not peer reviewed. The package contains the PDF, its LaTeX source, the scripts and result files it cites, and the dated literature-search log behind its priority statements.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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