Dickman多项式及其常数
Dickman polylogarithms and their constants
摘要：
Dickman函数F（alpha）给出了大整数N没有超过N^alpha的素因子的渐进概率。它由精确递归L_k（alpha）=- int_alpha^{1/k} dx L_{k-1}（x/（1-x））/x定义的广义多项式的有限和给出，其中L_0（alpha）=1。当α趋于0时，这些Dickman多项式的行为定义了一系列有趣的常数C_k。我猜想exp（gamma z）/Gamma（1-z）是sum_{k\ge0} C_k z^k的母函数。对于整数k<11，我得到了F（1/k）的高精度估计，并将Dickman问题与凝聚态物理和量子场论中的问题进行了比较。
英文摘要：
The Dickman function F(alpha) gives the asymptotic probability that a large integer N has no prime divisor exceeding N^alpha. It is given by a finite sum of generalized polylogarithms defined by the exquisite recursion L_k(alpha)=- int_alpha^{1/k} dx L_{k-1}(x/(1-x))/x with L_0(alpha)=1. The behaviour of these Dickman polylogarithms as alpha tends to 0 defines an intriguing series of constants, C_k. I conjecture that exp(gamma z)/Gamma(1-z) is the generating function for sum_{k\ge0} C_k z^k. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
