Asymptotic error terms in Bonse-type inequalities

TL;DR

This paper confirms a conjecture related to Bonse-type inequalities by analyzing the asymptotic behavior of error terms, providing precise thresholds for their positivity, and establishing bounds under various hypotheses.

Contribution

It proves the conjecture about the inequality for x=0.1, derives the asymptotic expansion of the error term, and determines the minimal n for positivity, including bounds under the Riemann Hypothesis.

Findings

Confirmed the conjecture for x=0.1 at n=24,154,953

Derived asymptotic expansion showing positivity for large n when x > -2

Established effective bounds for the minimal n under unconditional and Riemann Hypothesis assumptions

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.