Boundedness of Riemann Zeta function

**EinStone** · March 23rd 2010, 04:33 PM

Show that $\frac{\zeta '(s)}{\zeta (s)} = -\sum_{n=1}^{\infty}\frac{\Lambda (n)}{n^s}$ is bounded in the right half plane Re s > 2.

( $\Lambda (n)$ is the Mangoldt Function)

**chiph588@** · March 23rd 2010, 06:53 PM

Originally Posted by EinStone

Show that $\frac{\zeta '(s)}{\zeta (s)} = -\sum_{n=1}^{\infty}\frac{\Lambda (n)}{n^s}$ is bounded in the right half plane Re s > 2.

( $\Lambda (n)$ is the Mangoldt Function)

Note that $\Lambda(n)\leq \log(n)\leq n$ .

Let $s=\sigma+it$

So $\left|\frac{\zeta '(s)}{\zeta (s)}\right| \leq \sum_{n=1}^\infty \left|\frac{\Lambda(n)}{n^s} \right| \leq \sum_{n=1}^\infty \left|\frac{n}{n^s} \right| = \sum_{n=1}^\infty \left|\frac{1}{n^{s-1}} \right| = \sum_{n=1}^\infty \frac{1}{n^{\sigma-1}} = \zeta(\sigma-1)$ .

Since $\zeta(\sigma-1)$ is bounded for $\sigma = \Re(s)>2$ , we are done.

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