필즈상 받은 알렝 콘느가 최근에 논문냈는데

Zeta Spectral Triples

We propose and investigate a strategy toward a proof of the Riemann Hypothesis based on a spectral realization of its non-trivial zeros. Our approach constructs self-adjoint operators obtained as rank-one perturbations of the spectral triple associated with the scaling operator on the interval $[λ^{-1}, λ]$. The construction only involves the Euler products over the primes $p \leq x = λ^2$ and produces self-adjoint operators whose spectra coincide, with striking numerical accuracy, with the lowest non-trivial zeros of $ζ(1/2 + i s)$, even for small values of $x$. The theoretical foundation rests on the framework introduced in "Spectral triples and zeta-cycles" (Enseign. Math. 69 (2023), no. 1-2, 93-148), together with the extension in "Quadratic Forms, Real Zeros and Echoes of the Spectral Action" (Commun. Math. Phys. (2025)) of the classical Caratheodory-Fejer theorem for Toeplitz matrices, which guarantees the necessary self-adjointness. Numerical experiments show that the spectra of the operators converge towards the zeros of $ζ(1/2 + i s)$ as the parameters $N, λ\to \infty$. A rigorous proof of this convergence would establish the Riemann Hypothesis. We further compute the regularized determinants of these operators and discuss the analytic role they play in controlling and potentially proving the above result by showing that, suitably normalized, they converge towards the Riemann $Ξ$ function.

arxiv.org

해당 연산자로 2부터 13까지의 소수로 50번째자리 리만제타영점을 0.001~10^{-55}의 엄청나게 작은 오차로 계산해냄

해당 연산자는 스펙트럼에서 위치-운동량이 제한된조건으로 시작해서 무한대로 보낼때 수렴하면(연산자가 positivity유지하면) 리만가설 증명완료랑 동치인데

강의도 듣고 논문도 계속 이해하는중인데 읽을수록 신기함

한발자국 남은거같다

- dc official App