论文标题
加权质量计数功能的渐近扩展
Asymptotic expansions of weighted prime power counting functions
论文作者
论文摘要
我们证明了$π（x）$，$π（x）$，$ \ operatorName {li}（x）$，$ \ operatorName {ri}（x）$和相关功能的几个渐近持续分数扩展\ frac {1} {k}π（\ sqrt [k] {x}）$是riemann Prime Counting函数，$ \ operatatorName {ri}（x）= \ sum_ {k = 1} \ operatatorName {li}（\ sqrt [k] {x}）$是riemann近似prime计数函数。我们还确定函数的渐近持续分数扩展$ \ sum_ {p \ leq x} p^s $，用于所有$ s \ in \ mathbb {c} $ at \ mathbb {c} $，带有$ \ operatoTorname {re}（re}（s）> -1 $ \ frac {1} {p} $和$ \ log \ prod_ {a^x <p \ leq a^{x+1}}}（1 -1/p）^{ - 1} $用于所有实际数字$ a> 1 $。我们还确定了函数$π（ax）-π（bx）$ for $ a> b> 0 $的渐近续分扩展的前几个术语。作为这些结果的推论，我们确定了这些功能的“线性化”阶段的最佳理性近似值。
We prove several asymptotic continued fraction expansions of $π(x)$, $Π(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $π(x)$ is the prime counting function, $Π(x) = \sum_{k = 1}^\infty \frac{1}{k}π(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ μ(k)}{k} \operatorname{li}(\sqrt[k]{x})$ 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 $π(ax)-π(bx)$ for $a > b > 0$. As a corollary of these results, we determine the best rational approximations of the "linearized" verions of these various functions.
