Asymptotic expansions of weighted prime power counting functions

TL;DR

This paper develops asymptotic continued fraction expansions for prime counting functions and related sums, providing new insights into their approximations and behavior for large inputs.

Contribution

It introduces novel asymptotic continued fraction expansions for prime counting functions and related sums, expanding understanding of their asymptotic behavior and rational approximations.

Findings

Derived continued fraction expansions for $\pi(x)$, $\Pi(x)$, $ ext{li}(x)$, and $ ext{Ri}(x)$.

Established asymptotic expansions for sums over primes and related functions for all relevant parameters.

Identified best rational approximations for linearized versions of these functions.

Abstract

We prove several asymptotic continued fraction expansions of $\pi(x)$, $\Pi(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $\pi(x)$ is the prime counting function, $\Pi(x) = \sum_{k = 1}^\infty \frac{1}{k}\pi(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ \mu(k)}{k} \operatorname{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function $\sum_{p \leq x} p^s$ for all $s \in \mathbb{C}$ with $\operatorname{Re}(s) > -1$, and of the functions $\sum_{a^x < p \leq a^{x+1}} \frac{1}{p}$ and $\log \prod_{a^x < p \leq a^{x+1}} (1 -1/p)^{-1}$ for all real numbers $a > 1$. We also determine the first few terms of an asymptotic continued fraction expansion of the function $\pi(ax)-\pi(bx)$ for $a > b > 0$. As a corollary of these results, we determine the best rational approximations of the "linearized" verions of these various functions.