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arXiv · 2004.01080

On the set $\{π(kn):\ k=1,2,3,\ldots\}$

Zhi-Wei Sun·Lilu Zhao
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Abstract

An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $π(kn)$ is prime, where $π(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive integer $n$ the set $\{π(kn):\ k=1,2,3,\ldots\}$ contains infinitely many $P_2$-numbers which are products of at most two primes. We also prove that under the Bateman--Horn conjecture the set $\{π(4k):\ k=1,2,3,\ldots\}$ contains infinitely many primes.

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Zhi-Wei Sun, Lilu Zhao. 2020-04-01. On the set $\{π(kn):\ k=1,2,3,\ldots\}$. https://arxiv.org/abs/2004.01080

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