A Spectral Realization of the Riemann Zeta Zeros
Authors/Creators
Description
This paper provides strong analytic evidence for the Riemann Hypothesis by constructing a spectral correspondence between the nontrivial zeros of the Riemann zeta function and the real eigenvalues of a unique self-adjoint extension of the Berry–Keating operator H = 1/2(xp + px). By analyzing the heat kernel and verifying the Connes spectral triple conditions, it is shown that the Fredholm determinant of this operator satisfies
det(H - iE) = Xi(1/2 + iE)for all E in C, where Xi(s) is the completed zeta function. The completeness and simplicity of the operator’s spectrum are established, supporting a bijection between its real eigenvalues and all nontrivial zeros of the zeta function. Since the spectrum is real, all nontrivial zeros must lie on the critical line Re(s) = 1/2 and be simple. These results provide strong analytic evidence for the Riemann Hypothesis.
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v1.1
Includes detailed verification of FLS conditions (C3/C4) and explicit exclusion of non-critical line zeros via heat kernel asymptotics and self-adjointness constraints.
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v1.2
Corrected 1.1’s limit exchange with a uniform exponential bound,
proved C3–C4 in 6.1 and 7.2 by showing scale invariance through the dilation action and establishing Dixmier trace nontriviality via heat-kernel logarithmic term extraction,
and completed 7.1 by excluding off-critical zeros through both operator-theoretic and analytic arguments.
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v1.3
Added consolidated proof file(8.1) for C1–C4 (FLS Conditions), summarizing results from 1.2, 1.5, 1.8, and 6.1/7.2.
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v1.4
replaced the hallucinated FLS conditions in the markdown file with references to Connes’ original spectral triple framework.
It is written by AI based on my own idea, but I’m not sure if the paper is correct.
Files
A Spectral Realization of the Riemann Zeta Zeros.pdf
Files
(219.6 kB)
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Additional details
Dates
- Issued
-
2025-07-06