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

Alfredo Sotelo Pejerrey

Faculty of Engineering

Research output: Contribution to journal › Original Article › peer-review

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.

Translated title of the contribution	Trazas de ciertos operadores integrales relacionado a la hipótesis de Riemann
Original language	American English
Article number	13
Pages (from-to)	24971
Number of pages	24983
Journal	AIMS Mathematics
Volume	8
Issue number	10
DOIs	https://doi.org/10.3934/math.20231274
State	Published - 2023

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Cite this

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title = "Traces of certain integral operators related to the Riemann hypothesis",

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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T1 - Traces of certain integral operators related to the Riemann hypothesis

AU - Sotelo Pejerrey, Alfredo

N1 - -

PY - 2023

Y1 - 2023

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

UR - https://www.aimspress.com/article/doi/10.3934/math.20231274

U2 - 10.3934/math.20231274

DO - 10.3934/math.20231274

M3 - Original Article

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SP - 24971

JO - AIMS Mathematics

JF - AIMS Mathematics

IS - 10

M1 - 13

ER -