Analytic continuation of the Riemann zeta function to the right half-plane

The identity $\zeta (s)=\eta (s)/(1-2^{1-s})$ continues the Riemann zeta function meromorphically to $\Re s>0$. Apparent singularities at nonreal zeros of $1-2^{1-s}$ are removable, as one sees by replacing $2$ with an integer $k$ for which $1-k^{1-s}\ne 0$. At $s=1$, $\eta (1)=\log 2$ and $1-2^{1-s}\sim (s-1)\log 2$, so $\zeta$ has a simple pole of residue one.