# convergence of Riemann zeta series

The series

 $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ (1)

converges absolutely for all $s$ with real part greater than 1.

Proof. Let  $s=\sigma+it$  where  $\sigma$ and $t$ are real numbers and  $\sigma>1$.  Then

 $\left|\frac{1}{n^{s}}\right|=\frac{1}{|e^{s\log{n}}|}=\frac{1}{e^{\sigma\log{n% }}}=\frac{1}{n^{\sigma}}.$

Since the series  $\sum_{n=1}^{\infty}\frac{1}{n^{\sigma}}$ converges, by the $p$-test (http://planetmath.org/PTest), for  $\sigma>1$, we conclude that the series (1) is absolutely convergent in the half-plane  $\sigma>1$.

Title convergence of Riemann zeta series ConvergenceOfRiemannZetaSeries 2015-08-22 13:15:14 2015-08-22 13:15:14 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 11M06 msc 30A99 ModulusOfComplexNumber ComplexExponentialFunction