Lemma 3.2 (Polynomial growth of $eta$ in half-planes).

• (i)
  $f(\zeta )$ extends to a meromorphic function on $ℂ$, with the only pole being a simple pole which is at $s=1$.
• (ii)
  Let $k\ge 0$ be an integer. Then, for $\mathrm{Re}(s)\ge -k$ and $|s-1|\ge \frac{1}{10}$ we have $|\zeta (s)|{\ll }_k|s{|}^{k+2}+1$.