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This work aims to provide a proof of the Riemann Hypothesis, a long-standing problem in mathematics.

July 6, 2026Open Access

A proof of the Riemann Hypothesis

Key Points

This work aims to provide a proof of the Riemann Hypothesis, a long-standing problem in mathematics.
Utilized the Weil quadratic form for ζ(s) on compactly supported smooth functions.
Applied Connes' semilocal trace formula to connect the quadratic form with a Hilbert-space norm.
Executed exact stabilization to fully recover the Weil form.
The quadratic form for ζ(s) was shown to be non-negative on C_c^∞(R_+^×), fulfilling Weil's criterion for the Riemann Hypothesis.

Abstract

Riemann stated the hypothesis in 1859. It has been open since. This paper proves it. The proof shows that the Weil quadratic form for ζ (s) is non-negative on Cc^∞ (R_+^×). By Weil's criterion this is the Riemann Hypothesis. The argument uses Connes' semilocal trace formula to realize the cutoff quadratic form as a Hilbert-space norm; exact stabilization then recovers the full Weil form.

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Cite This Study

Simon Velez (Sat,) studied this question.

synapsesocial.com/papers/6a4b453d997070ff83b5b21b https://doi.org/https://doi.org/10.5281/zenodo.21184595

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