A variant of the prime number theorem

Liu, Kui; Wu, Jie; Yang, Zhousheng

doi:10.48550/arxiv.2105.10844

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…In this section, we shall cite three lemmas, which will be needed in the next section. The first one is [5,Proposition 3.1].…”

Section: Preliminary Lemmasmentioning

confidence: 99%

On some sums involving the integral part function

Liu,

Wu,

Yang

2023

Int. J. Number Theory

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Denote by [Formula: see text], [Formula: see text] and [Formula: see text] the number of representations of [Formula: see text] as a product of [Formula: see text] natural numbers, the number of distinct prime factors of [Formula: see text] and the characteristic function of the square-free integers, respectively. Let [Formula: see text] be the integral part of real number [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] for [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

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“…In this section, we shall cite three lemmas, which will be needed in the next section. The first one is [5,Proposition 3.1].…”

Section: Preliminary Lemmasmentioning

confidence: 99%

On some sums involving the integral part function

Liu,

Wu,

Yang

2023

Int. J. Number Theory

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“…Subsequently, by using a result of Baker on 2-dimensional exponential sums [1, Theorem 6], Bordellès [1] sharpened the exponents in (1.1)-(1.2) and also studied some new examples. Very recently, Liu, Wu and Yang used the multiple exponential sums to derive better results than [1] (see [5,6]).…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by [5,6], we would like to improve (1.5) by using three-dimensional exponential sums. Noticing that four functions σ, ϕ, β and Ψ share many similarities such as multiplicative structures and rates of growth (see [7]), we can get a more general result.…”

Section: Introductionmentioning

confidence: 99%

On a sum involving certain arithmetic functions and the integral part function

Ma¹,

Sun²

2021

Preprint

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On some sums involving the integral part function

Liu,

Wu,

Yang

2021

Preprint

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Denote by τ k (n), ω(n) and µ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [t] be the integral part of real number t. For f = ω, 2 ω , µ 2 , τ k , we prove that10k−1 and ε > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès.

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A variant of the prime number theorem

Abstract: Let Λ(n) be the von Mangoldt function, and let [t] be the integral part of real number t. In this note, we prove that for any ε > 0 the asymptotic formulaholds. This improves a recent result of Bordellès, which requires 97 203 in place of 9 19 .

Cited by 3 publications

References 16 publications

On some sums involving the integral part function

On some sums involving the integral part function

On a sum involving certain arithmetic functions and the integral part function

On some sums involving the integral part function

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