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convergence of Riemann zeta series

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

(1)

with real part greater than 1.

Proof. Let $s = \sigma+it$ where $\sigma,\,t \in \mathbb{R}$ and $\sigma > 1$ . Then

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

Since the series $\sum_{n=1}^\infty\frac{1}{n^\sigma}$ converges, by the

-test, for $\sigma > 1$ , we conclude that the series (1) is absolutely convergent in the half-plane $\sigma > 1$ .

"convergence of Riemann zeta series" is owned by pahio.

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Riemann zeta function

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Cross-references: absolutely convergent, converges, real part, converges absolutely, series
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This is version 4 of convergence of Riemann zeta series, born on 2007-08-11, modified 2007-08-12.
Object id is 9853, canonical name is ConvergenceOfRiemannZetaSeries.
Accessed 550 times total.

Classification:

AMS MSC:	11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
	30A99 (Functions of a complex variable :: General properties :: Miscellaneous)

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