2010/04/04 by David Broadhurst, Broadhurst, David · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #Mathematical Physics (math-ph) #Other Condensed Matter (cond-mat.other) #cond-mat.other #hep-ph #math-ph #math.CA #math.MP
paper · pdf · doi:10.48550/arxiv.1004.0519
11 pages, LaTeX
arxiv created 2010/04/04 · arxiv updated 2010/04/07
The Dickman function F(alpha) gives the asymptotic probability that a large integer N has no prime divisor exceeding Nalpha. It is given by a finite sum of generalized polylogarithms defined by the exquisite recursion Lk(alpha)=- intalpha1/k dx Lk-1(x/(1-x))/x with L0(alpha)=1. The behaviour of these Dickman polylogarithms as alpha tends to 0 defines an intriguing series of constants, Ck. I conjecture that exp(gamma z)/Gamma(1-z) is the generating function for sumk≥0 Ck zk. I obtain high-precision evaluations of F(1/k), for integers k<11, and compare the Dickman problem with problems in condensed matter physics and quantum field theory.