Analytic continuation:
For \(\Re(s) > 0\): \(\zeta(s) = \eta(s)/(1-2^{1-s})\) where \(\eta(s)=\sum(-1)^{n-1}n^{-s}\).
For \(\Re(s) \le 0\): functional equation \(\zeta(s)=2^{s}\pi^{s-1}\sin(\tfrac{\pi s}{2})\Gamma(1-s)\zeta(1-s)\).
Convergence regions:
The Dirichlet series \(S_N(s)=\sum_{n=1}^{N}n^{-s}\) converges to \(\zeta(s)\) only for \(\Re(s) > 1\).
The prime zeta \(P(s)=\sum_{p}p^{-s}\) also requires \(\Re(s) > 1\).
Key features:
Simple pole at \(s=1\). Critical line at \(\Re(s)=\tfrac{1}{2}\) where non-trivial zeros lie (Riemann hypothesis).
Trivial zeros at \(s=-2,-4,-6,\ldots\)
Visualization:
Height shows \(\log_{10}|f(s)|\) (zeros appear as valleys, pole as peak).
Color encodes complex phase \(\arg(f(s))\).