  A well-known result, due to Dirichlet and later generalized by de la
Vallee-Poussin, expresses a relationship between the sum of fractional parts
and the Euler-Mascheroni constant. In this paper, we prove an asymptotic
relationship between the summation of the products of fractional parts with
powers of integers on one hand, and the values of the Riemann zeta function, on
the other hand. Dirichlet's classical result falls as a particular case of this
more general theorem.
