A generalization of Halász's theorem to Beurling's generalized integers
and its application

Zhang, Wen-Bin

doi:10.1215/ijm/1256063576

Cited by 15 publications

(10 citation statements)

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“…I proposed some years back the following conjecture: Such a conjecture is trivially true under the Riemann Hypothesis. In this respect, we note that [23] proves that in case of the Beurling's generalized integers, one can have M P (x) = o(x) without having ψ(x) ∼ x. This reference has been kindly shown to me by Harold Diamond whom I warmly thank here.…”

Section: Introductionmentioning

confidence: 74%

From explicit estimates for primes to explicit estimates for the Möbius function

Ramaré

2013

Acta Arith.

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Section: Introductionmentioning

confidence: 74%

From explicit estimates for primes to explicit estimates for the Möbius function

Ramaré

2013

Acta Arith.

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“…[6], [9,Chap. 14], [16]. Here we study the Beurling version of the assertion M(x) = o(x) and show this can be deduced (a) from the PNT under an Oboundedness condition N(x) = O(x) or (b) from a Chebyshev-type upper bound assuming N(x) ∼ ax (for a > 0) and the integral condition (3.2) (below).…”

Section: Introductionmentioning

confidence: 94%

“…Our method is inspired by W.-B. Zhang's proof of a Halász-type theorem for Beurling primes [16]. Our first step is to replace M(x) = o(x) by an equivalent asymptotic relation.…”

Section: Chebyshev Hypothesismentioning

confidence: 99%

M(x)=o(x) Estimates for Beurling numbers

Debruyne¹,

Diamond²,

Vindas³

2018

J. Théor. Nombres Bordeaux

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In classical prime number theory there are several asymptotic formulas said to be "equivalent" to the PNT. One is the bound M (x) = o(x) for the sum function of the Moebius function. For Beurling generalized numbers, this estimate is not an unconditional consequence of the PNT. Here we give two conditions that yield the Beurling version of the M (x) bound, and examples illustrating failures when these conditions are not satisfied.

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“…A function f : N −→ C is said to be multiplicative on N if f (1) = 1 and f (mn) = f (m)f (n) whenever (m, n) = 1. Such an f is said to be completely multiplicative [5,16] if we also have f (mn) = f (m)f (n) for all values of m, n ∈ N , where (m, n) is defined as the largest ginteger that divides both m and n. We define the generalised Liouville function on N , is an example of completely multiplicative function, to be λ P (1) = 1 and λ P (n) = (−1)…”

Section: Completely Multiplicative Function On Nmentioning

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 generalization of Halász's theorem to Beurling's generalized integers and its application

Cited by 15 publications

References 0 publications

From explicit estimates for primes to explicit estimates for the Möbius function

From explicit estimates for primes to explicit estimates for the Möbius function

M(x)=o(x) Estimates for Beurling numbers

Completely multiplicative functions with sum zero over generalised prime systems

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