  The author derives new family of series representations for the values of the
Riemann Zeta function $\zeta(s)$ at positive odd integers. For
$n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges
remarkably rapidly with its general term having the order estimate:
$$O(m^{-2k}\cdot k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).$$ The
method is based on the mapping relationships between analytic functions and
periodic functions using the abstract operators $\cos(h\partial_x)$ and
$\sin(h\partial_x)$, including the mapping relationships between power series
and trigonometric series, if each coefficient of a power series is respectively
equal to that of a trigonometric series. Thus we obtain a general method to
find the sum of the Dirichlet series of integer variables. By defining the Zeta
function in an abstract operators form, we have further generalized these
results on the whole complex plane.
