The orbifold Euler characteristic of the moduli space \({{\cal M}_{g;1}}\) of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\), where ζ is the Riemann zeta function and B_2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.

中文翻译：

---

计算曲线模空间的欧拉特性

Harer 和 Zagier 计算了具有一个标记点​​ ( g ≥ 1)的属g光滑曲线的模空间\({{\cal M}_{g;1}}\)的 Orbifold Euler 特征： \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\)，其中 ζ 是黎曼 zeta 函数，B _{2 g}是第 (2 g ) 个伯努利数。我们仅使用形式幂级数和经典组合学对这个结果进行了更简短的证明。