P(s) = Σ p⁻ˢ over all primes. Visualise its values, partial sums, and convergence on the complex plane.
P(s) = Σ_{p prime} p⁻ˢ converges for Re(s)>1. Via Möbius inversion: P(s) = Σ_{n≥1} μ(n)/n · log ζ(ns). It has a logarithmic branch cut accumulating near Re(s)=1 from the zeros of ζ. On the real axis, P(2) = Σ1/p² ≈ 0.4522, and P(1)=+∞ (prime harmonic series diverges — Euler). The partial sums converge slowly; each new prime adds a tiny term.