Abstract analytic number theory is a branch of mathematics which takes the ideas and techniques of classical analytic number theory and applies them to a variety of different mathematical fields. The classical prime number theorem serves as a prototypical example, and the emphasis is on abstract asymptotic distribution results. The theory was invented and developed by mathematicians such as John Knopfmacher and Arne Beurling in the twentieth century. == Arithmetic semigroups == The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying t...

Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in (Hilbert 2013, p.730), and was popularized through George Gamow's 1947 book One Two Three... Infinity. == The paradox == Consider...