Inversion formula for the growth function of a cancellative monoid

TL;DR

This paper establishes an inversion formula linking the growth function and skew-growth function of cancellative monoids, generalizing the Euler product formula for the Riemann zeta function in a broader algebraic context.

Contribution

It introduces the skew-growth function and proves the inversion formula connecting it with the growth function for cancellative monoids.

Findings

Proves the inversion formula P(t)N(t)=1 for cancellative monoids.

Shows the connection to the Euler product formula for the Riemann zeta function.

Provides a new algebraic framework generalizing classical number theory results.

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.