Granville-Zagier の和公式の別証明

示したい等式の母関数をとる.
十分小さい正の実数 $A,W$ をとると,
$$\begin{array}{rl}\sum _{0<r<k}\sum _{\begin{array}{c}{k}_1,\dots ,k_r\ge 1\\ k_r\ge 2\\ k_1+\cdots +k_r=k\end{array}}\zeta (k_1,\dots ,k_r)A^r{W}^k& =\sum _{0<r<k}\sum _{\begin{array}{c}{k}_1,\dots ,k_r\ge 1\\ k_r\ge 2\\ k_1+\cdots +k_r=k\end{array}}(-1{)}^{r}I(0;1,\underset{k_1-1}{\underbrace{0,\dots ,0}},\dots ,1,\underset{k_r-1}{\underbrace{0,\dots ,0}};1)A^r{W}^k\\ & =\sum _{k=2}^{\mathrm{\infty }}\sum _{\begin{array}{c}{a}_1,\dots ,a_k\in \{0,1\}\\ a_1=1,a_k=0\end{array}}I(0;a_1,\dots ,a_k;1)(-A{)}^{a_1+\cdots +a_k}{W}^k\\ & =\sum _{k=2}^{\mathrm{\infty }}\sum _{\begin{array}{c}{a}_1,\dots ,a_k\in \{0,1\}\\ a_1=1,a_k=0\end{array}}{\int }_{0<s<t<1}\frac{-A}{s-1}I(s;a_2,\dots ,a_{k-1};t)\frac{1}{t}dsdt\cdot (-A{)}^{a_2+\cdots +a_{k-1}}{W}^k\\ & =\sum _{k=2}^{\mathrm{\infty }}{\int }_{0<s<t<1}\frac{-AW}{s-1}\cdot \frac{1}{(k-2)!}{\left({\int }_s^t(\frac{-AW}{x-1}+\frac{W}{x})dx\right)}^{k-2}\frac{W}{t}dsdt\\ & =-A{W}^2{\int }_{0<s<t<1}{s}^{-W}(s-1{)}^{AW-1}ds{t}^{W-1}(t-1{)}^{-AW}dt\end{array}$$
がいえる.
変数変換 ${\displaystyle (x,y)=(\frac{s(1-t)}{t(1-s)},\frac{1-t}{1-s})}$ を考える. このJacobianは
$$\left|\frac{\partial (s,t)}{\partial (x,y)}\right|=\frac{(y-x)(y-1)}{(x-1{)}^3{y}^2}$$
と計算できて, 積分範囲は $0<x<y<1$ にうつるため
$$\begin{array}{rl}\sum _{0<r<k}\sum _{\begin{array}{c}{k}_1,\dots ,k_r\ge 1\\ k_r\ge 2\\ k_1+\cdots +k_r=k\end{array}}\zeta (k_1,\dots ,k_r)A^r{W}^k& =A{W}^2{\int }_{0<x<y<1}\frac{x^{-W}}{x-1}dx{y}^{W-AW-1}dy\\ & =\frac{AW}{1-A}{\int }_0^1\frac{x^{-W}-x^{-AW}}{x-1}dx\\ & =\frac{AW}{1-A}\sum _{k\ge 2}\frac{W^{k-1}}{(k-1)!}(1-A^{k-1})I(x;0;1{)}^{k-1}\frac{dx}{x-1}\\ & =\sum _{k\ge 2}{\int }_0^1({\int }_{x<t_1<\cdots <t_{k-1}<1}\frac{d{t}_1}{t_1}\cdots \frac{d{t}_{k-1}}{t_{k-1}})\frac{dx}{x-1}\frac{A(1-A^{k-1})}{1-A}{W}^k\\ & =\sum _{k\ge 2}I(0;1,\underset{k-1}{\underbrace{0,\dots ,0}};1)\frac{A(1-A^{k-1}}{1-A}{W}^k\\ & =\sum _{k\ge 2}\zeta (k)(1+A+\cdots +A^{k-1})W^k\end{array}$$
を得る. 両辺の $A^r{W}^k$ の係数を比較すれば所望の等式を得る.