Heat Expansion and Zeta

2024/02/20 by Alain Connes, Connes, Alain · 1 citation
Mathematics · Physics and Astronomy · #11M06 #34B24 #41A60 #Advanced Optimization Algorithms Research #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.2402.13082

openalex publication_date 2024/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We compute the full asymptotic expansion of the heat kernel Trace(exp(-tD²)) where D is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The coefficients of the expansion are explicit expressions involving Bernoulli and Euler numbers. We relate the divergent terms with the heat kernel expansion of the Dirac square root of the prolate wave operator investigated in our joint work with Henri Moscovici.

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• The Riemann Hypothesis: Past, Present and a Letter Through Time

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