**On Adjusted Hilbert-Samuel Functions **

*Nguyen Tuan Long *

**Abstract**

Let *M* a finitely generated *R*-module of dimension *d*. Let *M*. Consider an adjusted Hilbert-Samuel function in *n* defined by

where *i*th arithmetic degree of *M* with respect to *n*_{0} such that *n*≥*n*_{0}. Moreover, if *M* is sequentially generalized Cohen-Macaulay, then *n*_{0} exists independently of the choice of