Exercise 15, § 9

Put N = Card(R$_+$)$,c=\prod_{\alpha\in R_+}\langle \rho , H_{\alpha}\rangle$, and $d(\lambda ) =$ dim $E(\lambda )$ if $\lambda \in P_{++}$.

Show that, for any real number $s >0$,

Deduce that $\sum_{\lambda\in P_{++}}d(\lambda )^{-s}<+\infty$ if $s >1$.