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 $E_n(z)$ correspondingly yield two asymptotic continued fraction expansions of $\pi(x)/x$. 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 $\pi(e^x)/e^x$. Finally, we generalize our results on $\pi(x)$ to any arithmetic semigroup satisfying Axiom A, and thus to any number field.