Inversion formula for the growth function of a cancellative monoid

If M=Z>0 is the multiplicative monoid of positive integers with deg=log, then PM,deg(exp(−s)) is Riemann zeta function and the inversion formula turns out to be the Euler product formula. We discuss more examples on the behavior of the analytic function NM,deg(exp(−s)) on the boundary line ℜ(s)=abscissa of absolute convergence of PM,deg(exp(−s)).