Continued-fraction expansions for the Riemann zeta function and polylogarithms期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Continued-fraction expansions for the Riemann zeta function and polylogarithms

Authors:	Djurdje Cvijovic Jacek Klinowski

Institution:	Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom ; Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom

Abstract:	It appears that the only known representations for the Riemann zeta function $\zeta (z)$ in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of $\zeta (n)$ for any integer $n\geq 2$ . This is a special case of a more general expansion which we have derived for the polylogarithms of order , $n\geq 1$ , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for $\log (1+z)$ . Computation demonstrates rapid convergence. For example, the 11th approximants for all $\zeta (n)$ , $n\geq 2$ , give values with an error of less than 10 $^{-9}$ .

Keywords:	Riemann zeta function polylogarithms continued fractions

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