A variant of the prime number theorem and some related problems
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].
Domaines
Théorie des nombres [math.NT]Origine | Fichiers produits par l'(les) auteur(s) |
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