Asymptotic error terms in Bonse-type inequalities

Abstract

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - \pi(n)}$ for all $n \geq 2$, where $\pi(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing the exponent $k(n,x) = n - \pi(n) + \frac{\pi(n)}{\pi(\log n)} - x \cdot \pi(\pi(n))$, proving the inequality holds for $x = 2$ and $n \geq 8$. In 2022, Marques and Trojovsk\'y extended this to $x = 1.4$ for $n \geq 21$ and conjectured its validity for $x = 0.1$ when $n \geq 24,154,953$. 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 $\Psi(x)$ such that $E_n(x) > 0$ for all $n \geq \Psi(x)$, precisely determining $\Psi(0.1) = 24,154,953$. Additionally, we establish effective upper bounds for $\Psi(x)$ both unconditionally and under the Riemann Hypothesis, with the conditional bounds showing a significant improvement. Our analysis fully resolves the conjecture and characterizes $\Psi(x)$ 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.