Asymptotic expansions of the prime counting function

Abstract

We provide several asymptotic expansions of the prime counting function π(x) and related functions. We define an asymptotic continued fraction expansion of a complex-valued function of a real or complex variable to be a possibly divergent continued fraction whose approximants provide an asymptotic expansion of the given function. We show that, for each positive integer n, two well-known continued fraction expansions of the exponential integral function En(z) correspondingly yield two asymptotic continued fraction expansions of π(x)/x. We prove this by first establishing some general results about asymptotic continued fraction expansions. We show, for instance, that the “best” rational function approximations of a function possessing an asymptotic Jacobi continued fraction expansion are precisely the approximants of the continued fraction, and as a corollary we determine all of the best rational function approximations of the function π(ex)/ex. Finally, we generalize our results on π(x) to any arithmetic semigroup satisfying Axiom A, and thus to any number field.