ON A REPRESENTATION OF THE RIEMANN ZETA FUNCTION

Abstract

In this paper a new representation of the Riemann function  in the disc $U(2,1)$ is obtained:$\zeta (z) = \dfrac{1}{z-1} + \displaystyle\sum_{n=0}^\infty(-1)^n\alpha_n(z-2)^n,$ where the coefficients $\alpha_k$ are real numbers tending to zero. Hence is obtained $\gamma=\displaystyle\lim_{m\rightarrow\infty}\left[\displaystyle\sum_{k=0}^n\dfrac{\zeta^{(k)}(2)}{k!}-n\right],$ where $\gamma$  is the Euler-Mascheroni constant.