多重 Zeta 函数值与 Euler 和式的研究

Abstract

在1742年回复哥德巴赫的信中，欧拉考虑了如下形式的二重和式 \[\sum\limits_{n=1}^\infty{\left({1+\frac{1}{{{2^m}}}+\cdots+\frac{1}{{{n^m}}}}\right)/{n^p}}，\] 其中$p$和$q$都为正整数且$q\geq2$.这种类型的和式后来就被称为线性的(或二重)欧拉和式.从这开始， 对$k$重欧拉和式封闭值的研究一直吸引着许多的专业和业余研究人员的兴趣，它是古典Riemannzeta函数的推广，一般有如下两种定义 \[\zeta\left({{s_1}，{s_2}，\cdots，{s_m}}\right...

In response to a letter from Goldbach in 1742, Euler considered sums of the form \[\sum\limits_{n = 1}^\infty {\left( {1 + \frac{1}{{{2^m}}} + \cdots + \frac{1}{{{n^m}}}} \right)/{n^p}}, \] where $p$ and $q$ are positive integers with $q\geq 2$. These kind of sums are called the linear (or double) Euler sums today. Historically, the k-fold Euler sums has attracted specialists and nonspeciali...