Multiplicative Functions

Number Theory · Axiom Academy

LESSON Multiplicative Functions Exploring the formal definitions, properties, and key examples of multiplicative and completely multiplicative functions in number theory An arithmetic function f : ℕ → ℂ is called multiplicative if: for all positive integers m and n with gcd( m , n ) = 1. Completely Multiplicative Function An arithmetic function f : ℕ → ℂ is called completely multiplicative if: for all positive integers m and n (no coprimality requirement). Key Distinction: Completely multiplicative functions satisfy the multiplicative property for all pairs of integers, while multiplicative functions only require it for coprime pairs. 2. Key Properties and Examples Since gcd(1, 1) = 1, we have f (1·1) = f (1)· f (1), which gives f (1) = [ f (1)]². Thus f (1)·[1 - f (1)] = 0, so f (1) = 0 or f (1) = 1. If f (1) = 0, then for any n with gcd( n , 1) = 1, we get f ( n ) = f ( n ·1) = f ( n )· f (1) = 0, making f identically zero. For non-trivial multiplicative functions, f (1) = 1. Multiplicative (but not completely): φ ( n ) - Euler's totient function (counts integers ≤ n coprime to n ) τ ( n ) or d ( n ) - number of divisors of n σ ( n ) - sum of divisors of n N ( n ) = n (identity function) n k for any fixed k (power functions) 1( n ) = 1 for all n (constant function) 3. Prime Factorization Property One of the most powerful aspects of multiplicative functions is that they are completely determined by their values at prime powers .

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