The maximal order of the shifted-prime divisor function

For each positive integer $n$, we denote by ${\omega }^{\ast }(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., $${\omega }^{\ast }(n):=\sum _{p-1\mid n}1.$$ 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 $c_0>0$, $${\omega }^{\ast }(n)>\exp \left(c_0\frac{\log n}{(\log \log n{)}^2}\right)$$ 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 $${\omega }^{\ast }(n)>\exp \left((\frac{1}{2}\log 2+o(1))\frac{\log n}{\log \log n}\right)$$ 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 $n$.