xvi The Main Functions

σ(n) =

d|n

d, the sum of the divisors of n

σ∗(n) =

d|n, (d,n/d)=1

d, the sum of the unitary divisors of n

σk(n) =

d|n

dk,

the sum of the

kth

powers of the divisors of n

σI (n) =

d|n,d

odd

d, the sum of the odd divisors of n

τ (n) =

d|n

1, the number of divisors of n

ω(n) =

p|n

1, the number of distinct prime factors of n

Ω(n) =

pα n

α, the number of prime factors of n counting their multiplicity

πk(x) =

n≤x

ω(n)=k

1, the number of numbers n ≤ x such that ω(n) = k

µ(n), the Moebius function defined by

µ(n) =

⎧

⎨

⎩

1 if n = 1,

0 if

p2|n

for a certain prime p,

(−1)ω(n)

otherwise

λ0(n) =

(−1)Ω(n),

the Liouville function

λ(n) =

log n

log γ(n)

, the index of composition of the number n ≥ 2

ι(n) = min

1≤m=n

P (m)≤P (n)

|n − m|, the index of isolation of the number n ≥ 2, that is the dis-

tance to the nearest integer whose largest prime factor does not exceed that of n

ξ(n) =

n

i=1

1

gcd(i, n)