The average order of the Möbius function for Beurling primes

Neamah, Ammar Ali; Hilberdink, Titus

doi:10.1142/s1793042120500517

Cited by 8 publications

(9 citation statements)

References 11 publications

(22 reference statements)

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“…In this section we simultaneously give a proof of Proposition 1. 4 and address an open question from [14]. In preparation, we need to introduce some concepts.…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

“…Following 5 Hilberdink and Neamah (cf. [14]), we define the three numbers α, β, γ as the unique exponents (necessarily elements of [0, 1]) for which the relations…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

“…We then call such a Beurling generalized number system an [α, β, γ]-system. The main result 6 of [14] (see also [7]) tells us that that Θ = max{α, β, γ} is at least 1/2 and that at least two of these numbers must be equal to Θ. Hilberdink and Lapidus [8] call a Beurling number system well-behaved…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

“…If the RH holds, then the rational integers are a [1/2, 0, 1/2]system, so that we have a candidate example of this instance. It is conjectured in [14] that there are no [1/2, 1/2, 0]-systems. The following open question is also posed in [14,Section 4]: Does there exist a [0, β, β] system with β < 1?…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

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A new generalized prime random approximation procedure and some of its applications

Broucke,

Vindas

2021

Preprint

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We present a new random approximation method that yields the existence of a discrete Beurling prime system P = {p 1 , p 2 , . . . } which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function F . This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704].We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed. 2020 Mathematics Subject Classification. 11M41, 11N80. Key words and phrases. Discrete random approximation; Diamond-Montgomery-Vorhauer-Zhang probabilistic method; Dirichlet series with unique zero in half plane of convergence; Well-behaved Beurling primes and integers; Beurling integers with large oscillation; Riemann hypothesis for Beurling numbers.F.

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“…In this section we simultaneously give a proof of Proposition 1. 4 and address an open question from [14]. In preparation, we need to introduce some concepts.…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

“…Following 5 Hilberdink and Neamah (cf. [14]), we define the three numbers α, β, γ as the unique exponents (necessarily elements of [0, 1]) for which the relations…”

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

Section: Some Examples Of Admissible Sequences Arementioning

confidence: 99%

See 2 more Smart Citations

A new generalized prime random approximation procedure and some of its applications

Broucke,

Vindas

2021

Preprint

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“…The concept of generalised primes and generalised integers was introduced by Beurling [3] in the 1930s and has been studied by many researchers since then (see for instance [2,4,11,15,18]). The structure of this system is defined to be a sequence of real positive numbers P = {p 1 , p 2 , p 3 , .…”

Section: Beurling Generalised Prime Systemsmentioning

confidence: 99%

Completely multiplicative functions with sum zero over generalised prime systems

Neamah

2020

Res. number theory

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CMO functions multiplicative functions f for which $$\sum _{n=1}^\infty f(n) =0$$ ∑ n = 1 ∞ f ( n ) = 0 . Such functions were first defined and studied by Kahane and Saïas [14]. We generalised these to Beurling prime systems with the aim to investigate the theory of the extended functions and we shall call them $$CMO_{\mathcal {P}}$$ C M O P functions. We give some properties and find examples of $$CMO_{\mathcal {P}}$$ C M O P functions. In particular, we explore how quickly the partial sum of these classes of functions tends to zero with different generalised prime systems. The findings of this paper may suggest that for all $$CMO_{\mathcal {P}}$$ C M O P functions f over $$\mathcal{{N}}$$ N with abscissa 1, we have $$\begin{aligned} \sum _{\tiny \begin{array}{c} n \le x \\ n\in \mathcal {N} \end{array}} f(n) = \Omega \Big (\frac{1}{\sqrt{x}} \Big ). \end{aligned}$$ ∑ n ≤ x n ∈ N f ( n ) = Ω ( 1 x ) .

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A new generalized prime random approximation procedure and some of its applications

Broucke,

Vindas

2024

Math. Z.

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The average order of the Möbius function for Beurling primes

Cited by 8 publications

References 11 publications

A new generalized prime random approximation procedure and some of its applications

A new generalized prime random approximation procedure and some of its applications

Completely multiplicative functions with sum zero over generalised prime systems

A new generalized prime random approximation procedure and some of its applications

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