Inversion formula for the growth function of a cancellative monoid

Abstract

We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge0}$ and associated generating function $P(t)=\sum_{m\in M}t^{deg(m)}$, called the growth function of $M$. We also introduce, using some towers of minimal common multiple sets in $M$, another signed generating function $N(t)$, called the skew-growth function of $M$. We show that these functions satisfy the inversion formula $P(t)N(t)=1$. In case the monoid is the set of positive integers with ordinary product structure and the degree map is logarithm function, using the coordinate change $t=exp(-s)$, the inversion formula turns out to be the Euler product formula for the Riemann's zeta function.