Exercise 2, § 8

We know (§ 6, Exercise 3) that if $n \geq 2$ the component of $H(U, V)$ of bidegree $(n, 1)$ is $\frac{1}{n!} b_n (\operatorname{ad} U)^n(V)$, where $b_n$ is the $n$-th Bernoulli number.

Deduce from this and Proposition 1 the inequality

Recover this result by means of the Clausen-von Staudt Theorem (Functions of a Real Variable, Chapter VI, § 2, Exercise 6) and show that there is equality if and only if $S(n) = p-1$.