Prime Number Theorem & Riemann Zeros

Prime Number Theorem (Hadamard & de la Vallée Poussin, 1896): π(x) ~ x/ln(x) as x→∞. The logarithmic integral Li(x) = ∫₂ˣ dt/ln(t) gives a much better approximation. Riemann's Explicit Formula: π(x) = Li(x) − Σᵨ Li(x^ρ) − ln(2) + ... where the sum is over all non-trivial zeros ρ of ζ(s). Each zero contributes an oscillatory correction. The Riemann Hypothesis (all zeros have Re(ρ)=1/2) is equivalent to π(x) = Li(x) + O(√x log x). The first few non-trivial zeros are ρ = 1/2 ± 14.135i, 1/2 ± 21.022i, ...