$F(n,N)$ has the form \begin{align} F(n,N) &= N \mu(a,b) + n[\, \xi(a,b,N) + b \log{n} \, ] \\ &= N \mu(a,b) + n b \log[{n \, e^{\xi/T}}] .\, \end{align} I know that $F(n,N)$ must be a homogeneous function of the first order in $n$ and $N$. From this condition one should be able to derive: $$ F(n,N) = N \mu(a,b) + n b \, \log \left[\frac{n}{N} \, g(a,b) \right],$$
with a new function $g(a,b)$. This means that $e^{\xi / T}$ is of the form $g/N$. For me this derivation is not clear; can somebody explain the missing insight?
From the definition of a homogeneous function of first order $$ F(wN,wn) = F(N,n) $$ we see that the logarithm is not a homogeneos function. To "make" the function homogeneous we must cancel the $w$ from the $n$-term in the argument of the logaritm, which is achieved by dividing by $w$ (therefore the $1/N$-term).