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
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