Dickman multiple polylogarithms and the Lindemann-Furry letters

Abstract

The Dickman function $\rho(u)$ gives the asymptotic probability that a large integer $N$ has no prime divisor exceeding $N^{1/u}$. We expand it in terms of rapidly computable multiple polylogarithms, as defined by Goncharov and intensively used for evaluations of Feynman integrals in quantum field theory. In parallel, we solve Buchstab's differential-delay equation, which concerns large integers $N$ divisible by no prime less than $N^{1/u}$. Discussion of the latter problem occurred in letters to the journal Nature during the second world war, from the physicists Frederick Lindemann and Wendell Furry. We recount how Furry evaluated a dilogarithm in reply to a puzzle resulting from Mertens' third theorem, raised by Lindemann. We refine Furry's analysis to include multiple polylogarithms of weights up to 200.