The maximal order of the shifted-prime divisor function

TL;DR

The paper advances the study of the maximal order of the shifted-prime divisor function $\omega^*(n)$, which counts primes $p$ with $p-1\mid n$, by sharpening Prachar's method and leveraging the anatomy of integers alongside probabilistic divisor constructions. It achieves explicit lower bounds on $\omega^*(n)$ for infinitely many $n$: unconditionally, $\omega^*(n) > \exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right)$, and under GRH, $\omega^*(n) > \exp\left((\log\frac{1+\sqrt{5}}{2}+o(1))\frac{\log n}{\log\log n}\right)$, both reflecting the maximal order growth. The arguments combine a refined Prachar framework with a probabilistic construction to maximize representations of the form $n=m(p-1)$, and they invoke Harman-type and zero-density estimates to control primes in arithmetic progressions, yielding explicit constants in the maximal order. The work also connects the maximal-order problem to the theory of smooth shifted primes, showing that the Adleman–Pomerance–Rumely conjecture would follow from fixed-θ prime-distribution hypotheses and, more broadly, tying these questions to conjectures on smooth numbers in shifted prime contexts with implications for Carmichael-type phenomena.

Abstract

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 $n$.

Theorems & Definitions (2)

• Theorem 1.1
• Proposition 3.1