Asymptotic expansions of weighted prime power counting functions

Abstract

We prove several asymptotic continued fraction expansions of $\pi (x)$, $\Pi (x)$, $\mathrm{li}(x)$, $\mathrm{Ri}(x)$, 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 $\mathrm{Ri}(x)={\sum }_{k=1}^{\infty }\frac{\mu (k)}{k}\mathrm{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function ${\sum }_{p\le x}{p}^s$ for all $s\in \mathbb{C}$ with $\mathrm{Re}(s)>-1$, and of the functions ${\sum }_{a^x<p\le a^{x+1}}\frac{1}{p}$ and $\log {\prod }_{a^x<p\le a^{x+1}}(1-1/p{)}^{-1}$ 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.

26 pages, 4 figures