The Riemann zeta function ζ(s) = Σ n-s converges only for Re(s) > 1, yet analytic continuation extends it to the entire complex plane. Zeta regularization assigns finite values to otherwise divergent sums: famously, ζ(−1) = −1/12, which appears in string theory and the Casimir effect. The critical strip 0 < Re(s) < 1 contains the non-trivial zeros; the Riemann Hypothesis conjectures all lie on Re(s) = 1/2. Partial sums are shown alongside the known analytic value — watch them diverge as Re(s) drops below 1, while the true ζ(s) remains finite.