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Given an arithmetic function $f:\mathbb{N} \to \mathbb{R}$, it is customaryto investigate the behavior of the corresponding sum ${\sum_{n\le N} f(n)}$ forlarge $N$. Here, for various classical arithmetic functions $f$, including thenumber of distinct prime factors function $\omega(n)$, Euler totient's function$\phi(n)$, the number of divisors function $d(n)$, the sum of the divisorsfunction $\sigma(n)$, as well as the middle divisors functions $\rho_1(n)$ and$\rho_2(n)$, we investigate the behavior of $f(n!)$ and their correspondingsums ${\sum_{n\le N} f(n!)}$. Finally, if $\lambda$ stands for the Liouvillefunction, according to the Chowla conjecture, ${\sum_{n\le N}\lambda(n)\lambda(n+1)=o(N)}$ as $N\to \infty$; here, we show that the analogueof the Chowla conjecture for factorial arguments is true as we prove that, as$N\to \infty$, we have  ${\sum_{n\le N} \lambda(n!)\lambda((n+1)!)=o(N)}$.

Sums of arithmetic functions running on factorials