Traces of certain integral operators related to the Riemann hypothesis

Alfredo Sotelo Pejerrey

doi:10.3934/math.20231274

Traces of certain integral operators related to the Riemann hypothesis

Título traducido de la contribución: Trazas de ciertos operadores integrales relacionado a la hipótesis de Riemann

Alfredo Sotelo Pejerrey

Facultad de ingeniería

Producción científica: Artículo Científico › Artículo original › revisión exhaustiva

Resumen

We prove the existence of a nontrivial singular trace τ defined on an ideal J closed with respect to the logarithmic submajorization such that τ(Aρ(α)) = 0, where Aρ(α) ∶ L 2(0, 1) → L 2(0, 1), [Aρ(α) f](θ) = ∫ 1 0 ρ(αθ/x) f (x)dx , 0 < α ≤ 1. We also show that τ(Aρ(α)) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr (Ar ρ (α)), r ∈ N, r ≥ 2.

Título traducido de la contribución	Trazas de ciertos operadores integrales relacionado a la hipótesis de Riemann
Idioma original	Inglés estadounidense
-	13
Páginas (desde-hasta)	24971
-	24983
Publicación	AIMS Mathematics
Volumen	8
N.º	10
DOI	https://doi.org/10.3934/math.20231274
Estado	Publicado - 2023

Nota bibliográfica

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Palabras clave

SINGULAR TRACE
SPECTRAL TRACE
modified Fredholm determinant
Riemann hypothesis

ODS de las Naciones Unidas

Este resultado contribuye a los siguientes Objetivos de Desarrollo Sostenible

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abstract = "We prove the existence of a nontrivial singular trace τ defined on an ideal J closed with respect to the logarithmic submajorization such that τ(Aρ(α)) = 0, where Aρ(α) ∶ L 2(0, 1) → L 2(0, 1), [Aρ(α) f](θ) = ∫ 1 0 ρ(αθ/x) f (x)dx , 0 < α ≤ 1. We also show that τ(Aρ(α)) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr (Ar ρ (α)), r ∈ N, r ≥ 2. ",

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AU - Sotelo Pejerrey, Alfredo

N1 - -

PY - 2023

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N2 - We prove the existence of a nontrivial singular trace τ defined on an ideal J closed with respect to the logarithmic submajorization such that τ(Aρ(α)) = 0, where Aρ(α) ∶ L 2(0, 1) → L 2(0, 1), [Aρ(α) f](θ) = ∫ 1 0 ρ(αθ/x) f (x)dx , 0 < α ≤ 1. We also show that τ(Aρ(α)) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr (Ar ρ (α)), r ∈ N, r ≥ 2.

AB - We prove the existence of a nontrivial singular trace τ defined on an ideal J closed with respect to the logarithmic submajorization such that τ(Aρ(α)) = 0, where Aρ(α) ∶ L 2(0, 1) → L 2(0, 1), [Aρ(α) f](θ) = ∫ 1 0 ρ(αθ/x) f (x)dx , 0 < α ≤ 1. We also show that τ(Aρ(α)) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr (Ar ρ (α)), r ∈ N, r ≥ 2.

KW - SINGULAR TRACE

KW - SPECTRAL TRACE

KW - modified Fredholm determinant

KW - Riemann hypothesis

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U2 - 10.3934/math.20231274

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M3 - Original Article

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JO - AIMS Mathematics

JF - AIMS Mathematics

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M1 - 13

ER -