Let $H(k)$ be the $k^{th}$ highly composite number (HCN).
- What is an asymptotic estimate for the size of $H(k)$ in terms of $k.$
- What is an asymptotic estimate for the size of $d(H(k))$ where $d(n)$ is the number of divisors of $n$.
Even though there are a lot of related questions and answers on Math Stackexchange, I haven't been able to find clear answers to these questions. Also, can one answer these questions for Superior HCNs?
Edit: A page at Bielefeld university here has numerical evidence suggesting $$ \ln H(k) \sim \ln 2 (\ln k)^{4/5}. $$ Has such a result been proved?