The maximum number of Hamiltonian cycles in tournaments

Let $C(n)$ denote the maximum number of directed Hamiltonian cycles in a tournament on $n$ vertices. We prove that, as $n\to\infty$, $$ C(n) = \left(\frac{\cosh 1}{\cos 1}\pm o(1)\right)\frac{(n-1)!}{2^n}. $$ The bound is asymptotically attained by the regular tournament with vertices $\mathbb Z/n