Divisor Functions τ and σ

Number Theory · Axiom Academy

LESSON Divisor Functions τ and σ Exploring the fundamental arithmetic functions that count and sum divisors Definition (Divisor Counting Function): For a positive integer n, the function τ(n) (also written d(n)) counts the total number of positive divisors of n. Definition (Divisor Sum Function): For a positive integer n, the function σ(n) sums all positive divisors of n. Watch the animation to see how we count and sum divisors for n = 12: Example: For n = 12, the divisors are 1, 2, 3, 4, 6, 12 Therefore: τ(12) = 6 and σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28 When n is a power of a prime p, the divisor functions have elegant closed-form expressions. Theorem (Prime Power Formulas): If n = p k where p is prime and k ≥ 1, then: The animation below demonstrates why these formulas work for p k = 2 4 = 16: Divisors are 1, 2, 4, 8, 16 = 2 0 , 2 1 , 2 2 , 2 3 , 2 4 σ(16) = (2 5 - 1)/(2 - 1) = 31/1 = 31 ✓ Both τ and σ are multiplicative functions , meaning they behave nicely under multiplication of coprime integers. This property allows us to compute τ(n) and σ(n) from the prime factorization: Watch how multiplicativity works for n = 12 = 2² × 3: τ(12) = τ(2²) × τ(3¹) = (2+1)(1+1) = 3 × 2 = 6 ✓ σ(12) = σ(2²) × σ(3¹) = [(2³-1)/(2-1)] × [(3²-1)/(3-1)] = 7 × 4 = 28 ✓ 4. Generalized Divisor Sum Function σₖ(n) The divisor sum function can be generalized to sum the k-th powers of divisors. Definition (Generalized Divisor Sum): For a positive integer n and non-negative integer k:

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