8 problems

Mundici's conjecture. The quantity $C(Q)$ is less than $3$ for all $Q\geq 2$.

For a non-negative integer $n$, the $n$th Farey sequence is … where the extended rationals $\widehat{\mathbb Q}$ consist of the usual rationals together with $1/0$. If…

For $x>1$, let the Farey series corresponding to $x$ be the ordered set of positive reduced rational numbers at most $1$ whose denominators are less than $x$. Let $A(x)$ be the num…

Shifted-correlation conjecture. For every $h\in{\mathbb N}^*$, there is a constant $A(h)$ such that

The vanishing total length conjecture. The total length tends to zero:

van Doorn's optimality conjecture. The constant $1/4$ in this upper bound is optimal, equivalently

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…