3 problems

Mundici's conjecture. One has

Let $p$ be a prime and let $n\geq 3$. A Farey sequence of $\Gamma_0(p^n)$ is denoted by $\{e_i\}$. Optimal upper-bound conjecture. The group $\Gamma_0(p^n)$ has a Farey sequence…

Let $\frac{a_1}{b_1},\frac{a_2}{b_2},\ldots$ be the Farey sequence of order $n$, and let $f(n)$ denote the least positive integer $r$ such that some pair of fractions $r$ positions…