Critical amplitude of the Potts model: Zeroes and divergences

The critical amplitude of the $q$-state Potts-model free energy is studied as a function of $q$ in two dimensions and on the diamond hierarchical lattice. The amplitude diverges at an infinite number of $q$ values, $q_n$, introducing logarithmic terms in the free energy. We expect that in each interval ($q_n,q_{n+1\mathrm{}}$) there is a value ${\hat{q}}_n$ where the amplitude vanishes, affecting the singularity of the free energy as a function of temperature. Possible consequences for gelation and vulcanization of polymers are discussed.