Asymptotic error terms in Bonse-type inequalities
Diego Marques, Pavel Trojovsky
TL;DR
The paper advances the understanding of Bonse-type inequalities by deriving an explicit asymptotic expansion for the logarithmic error term \\ E_n(x) \\) in the continuous Bonse framework, showing \\ E_n(x) = (x+2) \\frac{n}{\\log n} + O\\left(\\frac{n \\log\\log n}{\\log^2 n}\\right)\\ and establishing positivity for all sufficiently large \\ n \\ whenever \\ x>-2. It then characterizes the threshold function \\ Ψ(x) \\: for each \\ x>-2, identifying the minimal index from which the Bonse-type inequality holds; the paper proves \\ Ψ(0.1) = 24{,}154{,}953 \\ and provides both unconditional and RH-based upper bounds, with RH yielding substantially tighter bounds. A key structural result is that \\ Ψ(x) \\ is a non-increasing, right-continuous, piecewise-constant function whose jumps occur at a discrete set of thresholds \\ A_m \\, with \\ A_m = \,\sup_{n\ge m} \\alpha_n \\ and \\ E_n(x) = a_n + b_n x \\. The work also demonstrates the non-surjectivity of \\ Ψ \\, showing gaps of consecutive integers in its range, and provides an explicit RH-conditioned framework for improved thresholds. Overall, the results give a complete asymptotic description of the error term, certify the conjecture, and illuminate the fine-scale behavior of Bonse-type inequalities.
Abstract
Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - π(n)}$ for all $n \geq 2$, where $π(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing the exponent $k(n,x) = n - π(n) + \frac{π(n)}{π(\log n)} - x \cdot π(π(n))$, proving the inequality holds for $x = 2$ and $n \geq 8$. In 2022, Marques and Trojovský 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 $Ψ(x)$ such that $E_n(x) > 0$ for all $n \geq Ψ(x)$, precisely determining $Ψ(0.1) = 24,154,953$. Additionally, we establish effective upper bounds for $Ψ(x)$ both unconditionally and under the Riemann Hypothesis, with the conditional bounds showing a significant improvement. Our analysis fully resolves the conjecture and characterizes $Ψ(x)$ 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.