xviii Frequently Used Functions
γ(n) =
p|n
p, the product of the prime numbers dividing n
φ(n) =
m≤n
(m,n)=1
1, the Euler φ function
σ(n) =
d|n
d, the sum of the (positive) divisors of n
σk(n) =
d|n
dk,
the sum of the k-th powers of the divisors of n
d(n) =
d|n
1, the number of divisors of n
ω(n) =
p|n
1, the number of distinct prime divisors of n
Ω(n) =
pα n
α, the number of prime factors of n counting multiplicity
Πk(x) =
n≤x
ω(n)=k
1, the number of integers n ≤ x such that ω(n) = k
μ(n), the M¨ obius function, defined by
μ(n) =
⎧
⎨
⎩
1 if n = 1,
0
p2|n
for some prime p,
(−1)ω(n)
otherwise.
λ(n) =
log n
log γ(n)
, the index of composition of the integer n ≥ 2
