Let $K$ be a fixed positive integer and $X$ a discrete random variable with values in $\{1,2, \ldots, K\}$. Define the probability generating function of $X$. Express the mean of $X$ in terms of its probability generating function. The Dirichlet probability generating function of $X$ is defined as

$$q(z)=\sum_{n=1}^{K} \frac{1}{n^{z}} P(X=n)$$

Express the mean of $X$ and the mean of $\log X$ in terms of $q(z)$.