Asymptotic expansions of the prime counting function

TL;DR

This paper develops asymptotic continued fraction expansions for the prime counting function and related functions, providing new insights into their approximation and extending results to number fields.

Contribution

It introduces the concept of asymptotic continued fraction expansions and applies it to $\pi(x)$, connecting known continued fractions of $E_n(z)$ to asymptotic expansions of $rac{\pi(x)}{x}$, and generalizes to arithmetic semigroups.

Findings

Established general results on asymptotic continued fraction expansions.

Connected continued fraction approximants to best rational approximations of $rac{\pi(x)}{x}$.

Extended results to number fields satisfying Axiom A.

Abstract

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.