Pré-Publication, Document De Travail Année : 2024

A variant of the prime number theorem and some related problems

Bin Chen
  • Fonction : Auteur
  • PersonId : 1392025
Jiayuan Hu
  • Fonction : Auteur
  • PersonId : 1392026
Jie Wu

Résumé

Let $\Lambda(n)$ and $\mu(n)$ be the von Mangoldt function and the Möbius function. Let $\mathbb{1_P}(n)$ be the characteristic function of prime numbers and let $[t]$ be the integral part of real number $t$. In this paper, we prove that asymptotic formulas \begin{align*} \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) & = \bigg(\sum_{d=1}^{\infty} \frac{\Lambda(d)}{d(d+1)}\bigg) x + O_{\varepsilon}\big(x^{7/15+\varepsilon}\big) \\ \sum_{n\le x} \mu(n)^2 \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) & = \bigg(\frac{6}{\pi^2}\sum_{d=1}^{\infty} \frac{\Lambda(d)}{d(d+1)}\bigg) x + O\big(x^{2/3} \mathcal{L}(x)^{-c}\big) \end{align*} hold as $x\to\infty$, where $\mathcal{L}(x) := \exp((\log x)^{3/5}(\log\log x)^{-1/5})$, $\varepsilon>0$ is an arbitrarily small positive number and $c>0$ is a positive constant. The first asymptotic formula improves a recent result of Zhang, which requires $\frac{7}{15}+\frac{1}{195}$ in place of $\frac{7}{15}$. We also improve some results of Ma-Chen-Wu [13] and of Zhou-Feng [20].
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Dates et versions

hal-04611352 , version 1 (13-06-2024)

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  • HAL Id : hal-04611352 , version 1

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Bin Chen, Jiayuan Hu, Jie Wu. A variant of the prime number theorem and some related problems. 2024. ⟨hal-04611352⟩
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