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Asymptotic expansions of the prime counting function. (arXiv:1809.06633v1 [math.NT])
来源于:arXiv
We provide several new asymptotic expansions of the prime counting function
$\pi(x)$. 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
$E_n(z)$, in the regions where they diverge, correspondingly yield two
asymptotic continued fraction expansions of $\pi(x)/x$. We prove this by first
using Stieltjes' theory of moments to establish some general results about
Stieljtes and Jacobi continued fractions and then applying the theory
specifically to the probability measure on $[0,\infty)$ with density function
$\frac{t^n}{n!}e^{-t}$. We show generally that the "best" rational
approximations of a function possessing an asymptotic Jacobi continued fraction
expansion are precisely the a 查看全文>>