 Solution

A distribution on $\{1,2,\dots \}$ must have $\mu \ge 1$. For $\mu >1$, put $p=1/\mu$ and letFor any mass function $P$ with mean $\mu$, Gibbs inequality givesThus the geometric distribution uniquely maximizes entropy. For $\mu =1$, the only admissible law is the point mass at one, which is the limiting geometric case $p=1$.