Operator algebras associated with multiplicative convolutions of arithmetic functions

Aiju, Dong; Linzhe, Huang; Xue, Boqing

doi:10.1007/s11425-017-9281-0

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…Then the multiplicative monoid of operators L N = {L k : k ∈ N} shares the same structure with N. In particular, certain operator algebras generated by L N can reflect properties of natural numbers. The following results are shown in [13]: (i) The maximal ideal space of the Banach algebra generated by L N in B(l 2 (N)) is homeomorphic to D P . (ii) The C * -algebra generated by L N in B(l 2 (N)) contains no projections of finite rank other than 0.…”

Section: Riemann Hypothesis Holds If and Only Ifmentioning

confidence: 98%

“…In [13], Dong, Huang and the author considered the left regular action of N on l 2 (N). More concretely, for k ∈ N, let L k be the operator defined by…”

Section: Riemann Hypothesis Holds If and Only Ifmentioning

confidence: 99%

“…Also see [7][8][9] for some of the constructions related to Hilbert-Pólya conjecture. Moreover, a discrete-type 'differential operator' induced by a convolution of the Möbius function was studied in [13], which is also related to the Riemann ζ-function.…”

Section: Introductionmentioning

confidence: 99%

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On a Hilbert Space Reformulation of Riemann Hypothesis

Xue

2019

Preprint

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We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space H = A 2 1 (D), i.e., Riemann hypothesis holds if and only if the Hilbert subspace H 0 spanned by a certain family of functions coincides with H. A condition that a function does not belong to H ⊥ 0 is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid T N = {T k : k ∈ N} of operators is exactly B(H). As a result, Riemann hypothesis is true if and only if H 0 is T * k -invariant for all k ∈ N.

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Section: Riemann Hypothesis Holds If and Only Ifmentioning

confidence: 98%

“…In [13], Dong, Huang and the author considered the left regular action of N on l 2 (N). More concretely, for k ∈ N, let L k be the operator defined by…”

Section: Riemann Hypothesis Holds If and Only Ifmentioning

confidence: 99%

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On a Hilbert Space Reformulation of Riemann Hypothesis

Xue

2019

Preprint

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“…And commutative structures are often lifted to some corresponding non-commutative structures. In [6], the authors studied the multiplicative structure of natural numbers by operators and operator algebras through the left regular representation of N on l 2 (N). One of the theorems says that the C * -algebra generated by N in B(l 2 (N)) does not contain non-trivial projections of finite rank.…”

Section: Introductionmentioning

confidence: 99%

Natural Monoids and Non-commutative Arithmetics

Xue

2019

Preprint

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We introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid N of natural numbers in the group of positive rationals and 2) a certain monoid S in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics.Defining primes in a non-abelian monoid S is highly non-trivial, which relies on a concept we called "castling". Three types of castlings are essential to grasp the arithmetics on S. Multiplicative and completely multiplicative functions are defined. In particular, Möbius function is multiplicative, and Liouville function on a natural monoid is completely multiplicative. The divisor function has a sub-multiplicative property, which induces a non-trivial quantity τ 0 (u) = lim n→∞ (τ (u n )) 1/n in a non-abelian monoid S. Moreover, the quantity C ¸(S) = sup 1 =u∈S τ 0 (u)/τ (u) describes the complexity for castlings in S. We show that C ¸(N) = 1/2 and C ¸(S) = 1. The C * -algebra obtained from the left regular representation of S on l 2 (S), on which a particular trace can be defined, is also studied. Furthermore, we prove that a natural monoid having finitely many primes is amenable.

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Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

Gebremeskel

Huang

2019

Acta. Math. Sin.-English Ser.

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Operator algebras associated with multiplicative convolutions of arithmetic functions

Cited by 5 publications

References 13 publications

On a Hilbert Space Reformulation of Riemann Hypothesis

On a Hilbert Space Reformulation of Riemann Hypothesis

Natural Monoids and Non-commutative Arithmetics

Boundedness and Spectrum of Multiplicative Convolution Operators Induced by Arithmetic Functions

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