Frequently Used Functions P (n) = max{p : p|n}, the largest prime factor of n ≥ 2 p(n) = min{p : p|n}, the smallest prime factor of n ≥ 2 Ψ(x, y) = #{n ≤ x : P (n) ≤ y} Φ(x, y) = #{n ≤ x : p(n) y} β(n) = p|n p, the sum of the prime factors of n π(x) = p≤x 1, the number of prime numbers ≤ x θ(x) = p≤x log p ψ(x) = pα≤x α≥1 log p π(x k, ) = p≤x p≡ (mod k) 1, the number of prime numbers p ≤ x, p ≡ (mod k) li(x) = x 0 dt log t , the logarithmic integral xvii

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