Abstract: |
A growth function P_M(t) is a generating function for a degree map (valuation) defined on a monoid M. Its definition is simple, but it, often, turns out to be a meromorphic function in t, and its zero-loci and pole-loci contains interesting information: eg. Riemann's zeta function can be considered as the growth function for the monoid of all positive integers with ordinary product structure. The speaker came to this study in order to determine certain thermo-dynamical limit functions. In this talk, we discuss about the inversion formula for the growth function: in Riemann's zeta-function case, the inversion formula gives the Euler product structure. |