A variant of the prime number theorem

Kui Liu; Jie Wu; Zhishan Yang

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arXiv · 2105.10844

A variant of the prime number theorem

Kui Liu·Jie Wu·Zhishan Yang

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Abstract

Let $\Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $\varepsilon>0$ the asymptotic formula $$ \sum_{n\le x} \Lambda\Big(\Big[\frac{x}{n}\Big]\Big) = x\sum_{d\ge 1} \frac{\Lambda(d)}{d(d+1)} + O_{\varepsilon}\big(x^{9/19+\varepsilon}\big) \qquad (x\to\infty)$$ holds. This improves a recent result of Bordell\`es, which requires $\frac{97}{203}$ in place of $\frac{9}{19}$.

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Kui Liu, Jie Wu, Zhishan Yang. 2021-05-23. A variant of the prime number theorem. https://arxiv.org/abs/2105.10844

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