# Thread: Boundedness of Riemann Zeta function

1. ## Boundedness of Riemann Zeta function

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

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

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

($\displaystyle \Lambda (n)$ is the Mangoldt Function)
Note that $\displaystyle \Lambda(n)\leq \log(n)\leq n$.

Let $\displaystyle s=\sigma+it$

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

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