Asymptotic estimates for some number-theoretic power series
We study the asymptotic behaviour of the summatory function of a class of arithmetic functions. These functions are generalizations of the well-known general 4-dimensional divisor function d₄(n). We show that the corresponding error estimate is the best one can obtain by the present methods of analytic number theory.
The dynamical Mertens' theorem describes asymptotics for the growth in the number of closed orbits in a dynamical system. We construct families of ergodic automorphisms of fixed entropy on compact connected groups with a continuum of growth rates on two different growth scales. This shows in particular that the space of all ergodic algebraic dynamical systems modulo the equivalence of shared orbit-growth asymptotics is not countable. In contrast, for the equivalence relation of measurable isomorphism...