Spectral Flow for the Riemann zeros

2024/06/03 by André LeClair · 1 voice
Physics and Astronomy · Mathematics · #math-ph #cond-mat.dis-nn #hep-th #math.NT

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arxiv published 2024/06/03 · arxiv updated 2025/03/12

Abstract

Recently, with Mussardo we defined a quantum mechanical problem of a single particle scattering with impurities wherein the quantized energy levels E_n (σ) are exactly equal to the zeros of the Riemann ζ(s) where σ= \Re (s) in the limit σ→ 1/2. The S-matrix is based on the Euler product and is unitary by construction, thus the underlying hamiltonian is hermitian and all eigenvalues must be real. Motivated by the Hilbert-Pólya idea we study the spectral flows for \ E_n (σ) \. This leads to a simple criterion for the validity of the Riemann Hypothesis. The spectral flow arguments are simple enough that we present analogous results for the Generalized and Grand Riemann Hypotheses. We also illustrate our results for a counter example where the Riemann Hypothesis is violated since there is no underlying unitary S-matrix due to the lack of an Euler product.

Discussions

• 리만 제타함수의 근을 에너지 스펙트럼으로 갖는 물리계를 만들어서 그 물리계의 성질을 보는 것으로 리만 가설을 증명하려는 시도가 꽤 예전부터 있었는데 (Hilbert–Pólya 가설), 그 시도에 논문 하나가 더 추가. [bsky, 3 points, 1 comments]

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