Multiplicative Functions Magic

Number Theory · Axiom Academy

INTRO Multiplicative Functions Magic Discover how certain functions transform products into products! Let's start by exploring two numbers that share no common factors (except 1). Click on two numbers below to select them. Let's explore τ(n), which counts how many divisors a number has. Watch what happens when we compute it for your chosen numbers! Other Multiplicative Functions The divisor function isn't alone! Let's test whether other important number theory functions share this magic property. Let's visualize why the multiplicative property holds. Consider the divisors of m·n when m and n are coprime. When gcd(m, n) = 1, every divisor of m·n can be uniquely written as d₁·d₂ where d₁ divides m and d₂ divides n. A function f is multiplicative if f(1) = 1 and f(m·n) = f(m)·f(n) whenever gcd(m, n) = 1. τ(n): number of divisors σ(n): sum of divisors φ(n): Euler's totient function μ(n): Möbius function To compute f(n) for any n, just find the prime factorization n = p₁^a₁ · p₂^a₂ · ... · pₖ^aₖ, then: f(n) = f(p₁^a₁) · f(p₂^a₂) · ... · f(pₖ^aₖ)

This is the written version of the interactive lesson above. See the full Number Theory course.