On Huppert's $\rho$-$\sigma$ conjecture for primes in the orders of elements

Let $G$ be a finite group, let $\rho (G)$ denote the number of distinct prime divisors of $\mathrm{\mid }G\mathrm{\mid }$, and let $\sigma (G)$ be the maximal number of distinct prime divisors of the order of an element of $G$. Let $\rho (n)=\max \{\rho (G)\mid G\text{ solvable},\sigma (G)=n\}$. We prove that $\rho (n)\le 4n-1+\pi (2n)$ for all $n$, and, with the help of a computer, that $\rho (n)\le 9n\mathrm{/}2$ for all $n\le 16$. In particular, $\rho (n)\le \lfloor 9n\mathrm{/}2\rfloor +2$ and $\rho (n)\le (101\mathrm{/}22)n$ for all $n$, and $\rho (n)\le 9n\mathrm{/}2$ for all $n\ge 58$. This improves the bound $\rho (n)\le (17\mathrm{/}3)n$ due to Yang, and it shows that the bound $\rho (n)\le 5n$ stated by Bellotti, Keller and Trudgian holds, although its original proof contains a gap. Moreover, we show that $\rho (n)\ge 3n$ for all $n\ge 4$, which proves one half of a conjecture of the second author stating that $\rho (n)=3n$ for $n\ge 4$. We also derive a consequence for character codegrees and, conditionally, for arbitrary finite groups. The proofs of the main results do not rely on the classification of finite simple groups.