Inversion formula for the growth function of a cancellative monoid

Abstract

We consider any cancellative monoid $M$ equipped with a discrete degree map $deg:M\to R_{\ge 0}$ 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.

18 pages

Cited by in corpus (1)

• The skew growth functions $N_{M,\mathrm{deg}}(t)$ for the monoid of type ${\mathrm{B}}_{\mathrm{ii}}$ and others