Asymptotic expansions of the prime counting function
We provide several asymptotic expansions of the prime counting function \pi(x) and related functions. We define an \it 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 $E_n(z) in the regions where they diverge, correspondingly yield two asymptotic continued fraction expansions of \pi(x)/x$ We prove this by first using Stieltjes' theory of continued fractions to establish some general results about Stieljtes and Jacobi continued fractions and then applying the theory specifically to the probability measure on [0,\infty) with density function \frac{t^n}{n!}e^ t} We show generally 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 \pi(e^x)/e^x$ Finally, we generalize our results on \pi(x) to any arithmetic semigroup satisfying Axiom A.