Vol 83, No 5 (2019)

We study conformally invariant integral inequalities for real-valuedfunctions defined on domains $\Omega$ in $n$-dimensional Euclideanspace. The domains considered are of hyperbolic type, that is, they admita hyperbolic radius $R=R(x, \Omega)$ satisfying the Liouvillenon-linear differential equation and vanishing on the boundary of thedomain. We prove several inequalities which hold for all smooth compactlysupported functions $u$ defined on a given domain of hyperbolic type.Here are two of them:$$\int|\nabla u|^2R^{2-n}  dx \geq n (n-2)\int|u|^2R^{-n}  dx,$$$$\int|(\nabla u, \nabla R)|^p R^{p-s}  dx\geq \frac{2^pn^p}{p^p}\int|u|^pR^{-s}  dx,$$where $n\geq 2$, $1\leq p< \infty$ and $1+n/2 \leq s <\infty$.We also study the relations between Euclidean and hyperbolic characteristicsof domains.

We prove new theorems on the derivative of the Minkowski function.

We obtain an expression for the density of the distribution of the lengthsof arcs connecting neighbouring rational points on the unit circle withdenominators not exceeding a given bound.

An explicit formula is obtained expressing the Chebyshev psi-function in terms of the discrete spectrum of the Laplace operator on the fundamental domain of the modular group.