Lehmer's Interesting Series

2010/09/22 by F. J. Dyson, Dyson, F. J., N. E. Frankel +3 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #math-ph #math.MP #math.NT

paper · pdf · doi:10.48550/arxiv.1009.4274

14 pages

arxiv created 2011/03/28 · arxiv updated 2011/03/29

Abstract

The series S_k(z)=∑_m=1^∞\fracm^kz^m(\arrayc 2m m \array) is evaluated in non-recursive closed and analytically continued beyond its domain of convergence 0≤ |z|<4 for k=0,1,2,\.... From this we provide a firm basis for Lehmer's observation that π emerges from the limiting behavior of S_k(2) as k→∞.

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