Mathematics

On a representation of the Riemann zeta function

Yе. S. Mkrtchyan

Chair of Numerical Analysis and Mathematical Modeling YSU, Armenia

Abstract: In this paper a new representation of the Riemann zeta 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 $\gamma=\displaystyle\lim_{m\rightarrow\infty} \left[\displaystyle\sum_{k=0}^{n-1} \dfrac{\zeta^{(k)}(2)}{k!}-n\right]$, where $\gamma$ is the Euler–Mascheroni constant.

Keywords: Riemann function, Euler–Mascheroni constant, entire function, power series.

Received: 18.02.2016
Accepted: 06.06.2016

Language: English