RIMS - 1740 Inversion formula for the growth function of a cancellative monoid

Let (M, deg) be a cancellative monoid M equipped with a discrete degree map deg :M →R≥0, and let PM,deg(t) := ∑ u∈M/∼t deg(u) be its generating series, called the growth function of M . We show the inversion formula PM,deg(t) ·NM,deg(t) = 1 where the second factor of the formula is given by NM,deg(t) := 1 + ∑ T∈Tmcm(M,I0)(−1) #J1+···+#Jn−n+1 ∑ ∆∈|T | t , which we call the skew-growth function of M , and is a signed generating series of certain tree Tmcm(M, I0) of towers of minimal common multiple sets constructed inM of theminimalgenerator systemI0 :=min{M/∼\{1}}. If the monoid is (M, deg) = (Z>0, log), we get Riemann’s zeta function PZ>0,log(exp(−s))= ζ(s) as the growth function. Then the inversion formula turns out to be the Euler product formula ζ(s) ∏ p∈I0(1−p −s)=1.