Riemann zeta function

The Riemann zeta function is defined by the Dirichlet series

(1)   \begin{equation*}\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}\end{equation*}

where it converges absolutely for \Re(s)>1. By the application of the Euler-Maclaurin formula

(2)   \begin{equation*}\zeta(s)=\lim_{x\to\infty}\Big\{\sum_{n=1}^{x}\frac{1}{n^s}-\frac{x^{1-s}}{1-s}\Big\}\end{equation*}

extends the domain of converge for \Re(s)>1. And by subtracting one more term

(3)   \begin{equation*}\zeta(s)=\lim_{x\to\infty}\Big\{\sum_{n=1}^{x}\frac{1}{n^s}-\frac{x^{1-s}}{1-s}-\frac{1}{2x^s}\Big\}\end{equation*}

extends the domain of converge for \Re(s)>-1

Integral Representation

(4)   \begin{equation*}\zeta(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}dt\end{equation*}

where it converges absolutely for \Re(s)>1.