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

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