On the Khintchine Constant |
| |
Authors: | David H Bailey Jonathan M Borwein Richard E Crandall |
| |
Institution: | NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94035-1000 ; Centre for Experimental and Constructive Mathematics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada ; Center for Advanced Computation, Reed College, Portland, Oregon 97202 |
| |
Abstract: | We present rapidly converging series for the Khintchine constant and for general ``Khintchine means' of continued fractions. We show that each of these constants can be cast in terms of an efficient free-parameter series, each series involving values of the Riemann zeta function, rationals, and logarithms of rationals. We provide an alternative, polylogarithm series for the Khintchine constant and indicate means to accelerate such series. We discuss properties of some explicit continued fractions, constructing specific fractions that have limiting geometric mean equal to the Khintchine constant. We report numerical evaluations of such special numbers and of various Khintchine means. In particular, we used an optimized series and a collection of fast algorithms to evaluate the Khintchine constant to more than 7000 decimal places. |
| |
Keywords: | Khintchine constant continued fractions geometric mean harmonic mean computational number theory zeta functions polylogarithms |
|
| 点击此处可从《Mathematics of Computation》浏览原始摘要信息 |
| 点击此处可从《Mathematics of Computation》下载免费的PDF全文 |
|