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) =
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)
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