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Given a right LCM semigroup $S$ and a homomorphism $N\colon S\to[1,+\infty)$,we use the groupoid approach to study the KMS$_\beta$-states on $C^*(S)$ withrespect to the dynamics induced by $N$. We establish necessary and sufficientconditions for the existence and uniqueness of KMS$_\beta$-states. As anapplication, we show that the sufficient condition for the uniqueness obtainedfor so-called generalized scales is necessary as well. Our most completeresults are obtained for inverse temperatures $\beta$ at which the$\zeta$-function of $N$ is finite. In this case we get an explicit bijectivecorrespondence between the KMS$_\beta$-states on $C^*(S)$ and the tracialstates on $C^*(\operatorname{ker} N)$.

The groupoid approach to equilibrium states on right LCM semigroup C*-algebras