Asymptotic error terms in Bonse-type inequalities

Abstract

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1\cdots p_n>p_{n+1}^{n-\ddot{I}€(n)}$ for all $n\ge 2$, where $\ddot{I}€(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing the exponent $k(n,x)=n-\ddot{I}€(n)+\frac{\ddot{I}€(n)}{\ddot{I}€(\log n)}-x\cdot \ddot{I}€(\ddot{I}€(n))$, proving the inequality holds for $x=2$ and $n\ge 8$. In 2022, Marques and Trojovský extended this to $x=1.4$ for $n\ge 21$ and conjectured its validity for $x=0.1$ when $n\ge 24,154,953$. This paper confirms the conjecture by analyzing the error term $E_n(x)=\log (p_1\cdots p_n)-k(n,x)\log p_{n+1}$. Also, we derive the asymptotic expansion to $E_n(x)$ demonstrating that it is positive for all sufficiently large $n$ when $x>-2$. For each $x>-2$, we identify a minimal integer $\hat{I}¨(x)$ such that $E_n(x)>0$ for all $n\ge \hat{I}¨(x)$, precisely determining $\hat{I}¨(0.1)=24,154,953$. Additionally, we establish effective upper bounds for $\hat{I}¨(x)$ both unconditionally and under the Riemann Hypothesis, with the conditional bounds showing a significant improvement. Our analysis fully resolves the conjecture and characterizes $\hat{I}¨(x)$ as a non-increasing, piecewise constant function, exhibiting discontinuities at a discrete set of threshold points. These results advance the understanding of Bonse-type inequalities and their asymptotic behavior.