The Euler characteristic of $\operatorname{Out}(F_n)$

Understand

We prove that the rational Euler characteristic of $\operatorname{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma(n- \frac32)/\sqrt{2\pi} \log^2 n$.

• This settles a 1987 conjecture of J.
• Smillie and the second author.
• We establish connections with the Lambert $W$-function and the zeta function.