Completed Riemann Zeta Function has Order One

Theorem
The completed Riemann zeta function $\xi$ has order at most 1.

Proof
We are required to prove that


 * $\displaystyle \xi(s) = \frac 12 s(s-1) \pi^{-s/2} \Gamma\left(\frac s2\right) \zeta(s) \ll \exp\left({|s|^\beta}\right)$

for all $\beta > 1$, where $\ll$ is the order notation.

Note that by the Functional Equation for Riemann Zeta Function, it is sufficient to check this for $\Re(s) \geq 1/2$.

We simply check this fact for each factor.

Evidently

For some $c_1 > 0$.

For the gamma factor, we have Stirling's Formula for the Gamma Function


 * $\displaystyle \log \Gamma(s) = \left(s - \frac12\right)\log s - s + \frac{\log 2\pi}2 + \sum_{n = 1}^{d-1} \frac{B_{2n}}{2n(2n-1)s^{2n-1}} + \mathcal O\left( s^{1-2d} \right)$

This is valid only away from the poles of $\Gamma$ at $s = 0,-1,-2,\ldots$, but we assume $\Re(s) \geq 1/2$, so we are ok.

The error term $\mathcal O\left( s^{1-2d} \right)$ is small for large $s$.

More generally, the largest contribution is the term $\left(s - \frac12\right)\log s$, so we have


 * $\displaystyle \log \Gamma\left( \frac s2 \right) \ll |s| \log |s|$

That is,


 * $\displaystyle \Gamma\left( \frac s2 \right) \ll \exp(|s| \log |s|)$

Finally we have by Equivalence of Riemann Zeta Function Definitions, for $\Re(s) > 1/2$


 * $\displaystyle \zeta(s) = \frac s {s-1} - s \int_1^\infty \{ x\} x^{-s-1}\ dx$

We see that for $\Re(s) > 1/2$, the integral is bounded, and therefore


 * $(1-s) \zeta(s) \ll \mathcal O (|s|^2) \ll \exp(|s|)$

Combining these facts, and using that $\log s \ll s^\epsilon$ for all $\epsilon > 0$ (shown by Upper Bound of Natural Logarithm), we have


 * $\displaystyle |\xi(s)| \ll \exp\left( |s|^{1+ \epsilon} \right)$

for all $\epsilon > 0$, and we are done.