Analytic continuation

Analytic continuation is a way of extending a function beyond the region where its original formula directly converges, while preserving the analytic function determined there.

The defining series for $\zeta(s)$ converges when the real part of $s$ is greater than $1$. Analytic continuation extends the zeta function across almost the whole complex plane, which is what makes its critical strip and non-trivial zeros available for study.

See also: Riemann zeta function, complex plane, critical strip..