The UV prolate spectrum matches the zeros of zeta

Connes, Alain; Moscovici, Henri

doi:10.1073/pnas.2123174119

Cited by 11 publications

(19 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Alain Connes has found, starting around 1998, some intriguing connections between the so-called prolate spheroidal functions and zeros of the Riemann zeta function. This is well documented in references 1–3, 9, and 10 of the paper ( 1 ) commented upon here.…”

supporting

confidence: 79%

“…Ref. 1 adds another unexpected twist to the discoveries made so far. While previous work looked at the low zeros of the zeta function and their relationship to the eigenvalues of the traditional second-order differential operator in the interval [-λ, λ], in ref.…”

mentioning

confidence: 90%

“…While previous work looked at the low zeros of the zeta function and their relationship to the eigenvalues of the traditional second-order differential operator in the interval [-λ, λ], in ref. 1 , an appropriate self-adjoint extension of this operator to the real line is considered. This idea is already floating around in reference 9 of ref.…”

mentioning

confidence: 99%

“…This idea is already floating around in reference 9 of ref. 1 , but its full development is only accomplished here. There is no reason to suspect that this would have any connection with zeros of the zeta function, and yet…, surprise …, surprise …, it turns out that the negative eigenvalues of a self-adjoint operator in Hilbert space are related to the squares of the zeros of the zeta function.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Serendipity strikes again

Grünbaum

2022

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

supporting

confidence: 79%

mentioning

confidence: 90%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Serendipity strikes again

Grünbaum

2022

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

“…Remark 4.4. The main interest of the above reformulation of the spectral realization of [8,9] in terms of the Laplacian ∆ is that the latter is intimately related to the prolate wave operator W λ that is shown in [10] to be self-adjoint and have, for λ " ? 2 the same UV spectrum as the Riemann zeta function.…”

Section: The Laplacian ∆ " Hp1 `Hqmentioning

confidence: 99%

Hochschild homology, trace map and $ζ$-cycles

Connes¹,

Consani²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (i.e. non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator. The second spectral realization affects only the critical zeros and it is cast in terms of sheaf cohomology. The novelty is that the base space is the Scaling Site playing the role of the parameter space for the ζ-cycles and encoding their stability by coverings.

show abstract

Signal Communication and Modular Theory

Longo

2023

Commun. Math. Phys.

View full text Add to dashboard Cite

We propose a conceptual frame to interpret the prolate differential operator, which appears in Communication Theory, as an entropy operator; indeed, we write its expectation values as a sum of terms, each subject to an entropy reading by an embedding suggested by Quantum Field Theory. This adds meaning to the classical work by Slepian et al. on the problem of simultaneously concentrating a function and its Fourier transform, in particular to the “lucky accident” that the truncated Fourier transform commutes with the prolate operator. The key is the notion of entropy of a vector of a complex Hilbert space with respect to a real linear subspace, recently introduced by the author by means of the Tomita-Takesaki modular theory of von Neumann algebras. We consider a generalization of the prolate operator to the higher dimensional case and show that it admits a natural extension commuting with the truncated Fourier transform; this partly generalizes the one-dimensional result by Connes to the effect that there exists a natural selfadjoint extension to the full line commuting with the truncated Fourier transform.

show abstract

The UV prolate spectrum matches the zeros of zeta

Cited by 11 publications

References 25 publications

Serendipity strikes again

Serendipity strikes again

Hochschild homology, trace map and $ζ$-cycles

Signal Communication and Modular Theory

Contact Info

Product

Resources

About