Multiplicative Functions and Möbius Inversion Formula
نویسنده
چکیده
Examples. Let n ∈ N. Define functions τ, σ, π : N → N as follows: • τ(n) = the number of all natural divisors of n = #{d > 0 | d|n}; • σ(n) = the sum of all natural divisors of n = ∑ d|nd; • π(n) = the product of all natural divisors of n = ∏ d|nd. As we shall see below, τ and σ are multiplicative functions, while π is not. From now on we shall write the prime decomposition of n ∈ N as n = p1 1 p α2 2 · · · pαr r for distinct primes pi and αi > 1. It is easy to verify the following properties of multiplicative functions:
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