Problem Statement
A number $n$ is highly composite if $\tau(m)<\tau(n)$ for all $m<n$, where $\tau(m)$ counts the number of divisors of $m$. Let $Q(x)$ count the number of highly composite numbers in $[1,x]$.
Is it true that\[Q(x)\gg_k (\log x)^k\]for every $k\geq 1$?
Is it true that\[Q(x)\gg_k (\log x)^k\]for every $k\geq 1$?
Categories:
Number Theory Divisors
Progress
Erdős [Er44] proved $Q(x)\gg (\log x)^{1+c}$ for some constant $c>0$.The answer to this problem is no: Nicolas [Ni71] proved that\[Q(x) \ll (\log x)^{O(1)}.\]
Source: erdosproblems.com/381 | Last verified: January 14, 2026