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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.13691 |
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| _version_ | 1866914161510842368 |
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| author | Marques, Diego Trojovsky, Pavel |
| author_facet | Marques, Diego Trojovsky, Pavel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic error terms in Bonse-type inequalities Marques, Diego Trojovsky, Pavel Number Theory 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. |
| title | Asymptotic error terms in Bonse-type inequalities |
| topic | Number Theory |
| url | https://arxiv.org/abs/2511.13691 |