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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.14167 |
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| _version_ | 1866914096165683200 |
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| author | Fan, Steve Pollack, Paul |
| author_facet | Fan, Steve Pollack, Paul |
| contents | For each positive integer $n$, we denote by $ω^*(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., \[ω^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, this function has interesting applications in primality testing and bears a strong connection with counting Carmichael numbers. Prachar showed that for a certain constant $c_0 > 0$, \[ω^*(n)>\exp\left(c_0\frac{\log n}{(\log\log n)^2}\right)\] for infinitely many $n$. This result was later improved by Adleman, Pomerance and Rumely, who established an inequality of the same shape with $(\log\log n)^2$ replaced by $\log\log n$. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, Prachar also proved the stronger inequality \[ω^*(n)>\exp\left(\left(\frac{1}{2}\log2+o(1)\right)\frac{\log n}{\log\log n}\right)\] for infinitely many $n$. By refining the arguments of Prachar and of Adleman, Pomerance and Rumely, we improve on their results by establishing \begin{align*} ω^*(n)&>\exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right) \quad\text{(unconditionally)},\\ ω^*(n)&>\exp\left(\left(\log\left(\frac{1+\sqrt{5}}{2}\right)+o(1)\right)\frac{\log n}{\log\log n}\right) \quad\text{(under GRH)}, \end{align*} for infinitely many $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_14167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The maximal order of the shifted-prime divisor function Fan, Steve Pollack, Paul Number Theory For each positive integer $n$, we denote by $ω^*(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., \[ω^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, this function has interesting applications in primality testing and bears a strong connection with counting Carmichael numbers. Prachar showed that for a certain constant $c_0 > 0$, \[ω^*(n)>\exp\left(c_0\frac{\log n}{(\log\log n)^2}\right)\] for infinitely many $n$. This result was later improved by Adleman, Pomerance and Rumely, who established an inequality of the same shape with $(\log\log n)^2$ replaced by $\log\log n$. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, Prachar also proved the stronger inequality \[ω^*(n)>\exp\left(\left(\frac{1}{2}\log2+o(1)\right)\frac{\log n}{\log\log n}\right)\] for infinitely many $n$. By refining the arguments of Prachar and of Adleman, Pomerance and Rumely, we improve on their results by establishing \begin{align*} ω^*(n)&>\exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right) \quad\text{(unconditionally)},\\ ω^*(n)&>\exp\left(\left(\log\left(\frac{1+\sqrt{5}}{2}\right)+o(1)\right)\frac{\log n}{\log\log n}\right) \quad\text{(under GRH)}, \end{align*} for infinitely many $n$. |
| title | The maximal order of the shifted-prime divisor function |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.14167 |