Two Spectral Approaches to the Riemann Hypothesis: A Comparative Analysis of Connes' Adelic Framework and the EFM Operator

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This paper provides a comparative analysis of two spectral approaches to the Riemann Hypothesis. The first is Alain Connes' adelic framework (1999), which constructs a trace formula on the adele class space and proves that the Riemann Hypothesis is equivalent to the validity of a global trace formula. The S-local trace formula is proved; the global case remains open.

The second is the Euler-Fourier-Mellin (EFM) operator, an explicit operator built from prime shift operators on L²(ℝ⁺, dx/x). Its kernel on the critical line consists of Dirac deltas at the imaginary parts of zeros of ζ(1/2+it). A growth lemma is proved: functions of the form cosh(αu)e^{iγu} belong to the dual of the Gelfand-Shilov space S^{1/2}_{1/2}(ℝ) if and only if α=0.

Neither framework proves the Riemann Hypothesis. Both reduce it to an unproved statement. The paper identifies the relationship between the two approaches: the EFM operator corresponds to the Archimedean place in Connes' adelic construction. The gaps in both frameworks are precisely stated.

No claim of proof is made. This is a research program document.

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Keywords: Riemann Hypothesis, spectral theory, Connes trace formula, EFM operator, Gelfand-Shilov spaces, Arithmetic Spectral Theory

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1. Arithmetic Spectral Theory: A New Language for the Riemann Hypothesis: https://zenodo.org/records/19897850
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Category: Mathematics — Number Theory