Error estimates for a class of continuous Bonse-type inequalities

Abstract:

Let $p_n$ be the $n$th prime number. In 2000, Papaitopol proved that the inequality $p_1\cdots p_n>p_{n+1}^{n-\pi (n)}$ holds, for all $n\geq 2$, where $\pi (x)$ is the prime counting function. In 2021, Yang and Liao tried to sharpen this inequality by replacing $n-\pi (n)$ by $n-\pi (n)+\pi (n)/\pi (\log n)-2\pi (\pi (n))$, however there is a small mistake in their argument. In this paper, we exploit properties of the logarithm error term in inequalities of the type $p_1\cdots p_n>p_{n+1}^{k(n,x)}$, where $k(n,x)=n-\pi (n)+\pi (n)/\pi (\log n)-x\pi (\pi (n))$. In particular, we improve Yang and Liao estimate, by showing that the previous inequality at $x=1.4$ holds for all $n\geq 21$.