Is equality of the signed Euler characteristic of the moduli stack of elliptic curves and the sum of the divergent series 1+2+3+... a coincidence?

Question

(1) The signed Euler characteristic of the moduli stack of elliptic curves is $-1/12$.

(2) The Ramanujan-sum of the divergent series $1+2+3+\cdots$ Is also $-1/12$.

Is this a simple coincidence or can we use one equality to prove the other?

Answer 1

The following is taken form here

The orbifold Euler characteristic $\chi$ of $\mathcal{M}_{g,1}$ is given by the Riemann zeta function at negative integral values as follows (Zagier-Harer *): $\chi(\mathcal{M}_{g,1}) = \zeta(1-2g) \,$.

By the expression of the Riemann zeta function at negative integral values by the Bernoulli numbers $B_n,$ this says equivalently that $\chi(\mathcal{M}_{g,1}) = -\frac{B_{2g}}{2g} \,$.

For instance for $g = 1$ (once punctured complex tori, hence complex elliptic curves) this yields $\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}$ for the orbifold Euler characteristic of the moduli space of elliptic curves.

*Don Zagier, John Harer, The Euler characteristic of the moduli space of curves, Inventiones mathematicae (1986) Volume: 85, page 457-486 (EUDML)