Analytic continuation of the Riemann zeta function to the right half-plane
= Analytic continuation of the Riemann zeta function to the right half-plane
{c}
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 $1-k^{1-s}\ne0$. At $s=1$, $\eta(1)=\log2$ and $1-2^{1-s}\sim(s-1)\log2$, so $\zeta$ has a simple pole of residue one.