Tau and Sigma Functions: Counting and Summing Divisors

 

Two essential functions in number theory help us understand divisors: tau counts them, sigma adds them up!

The Tau Function τ(n)

τ(n) counts the total number of positive divisors of n.

Examples

τ(12) = 6 because divisors are {1, 2, 3, 4, 6, 12}

τ(7) = 2 (always 2 for primes!)


Formula: For n = p₁^a₁ · p₂^a₂ · ... · pₖ^aₖ:

τ(n) = (a₁ + 1)(a₂ + 1)...(aₖ + 1)

Example: τ(36) where 36 = 2² × 3²

τ(36) = (2 + 1)(2 + 1) = 9

The Sigma Function σ(n)

σ(n) gives the sum of all positive divisors of n.

Examples:

σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28

σ(6) = 1 + 2 + 3 + 6 = 12


Formula: For prime power p^k:

σ(p^k) = (p^(k+1) - 1)/(p - 1)

Both functions are multiplicative when factors are coprime!

Perfect Numbers

A number is perfect if σ(n) = 2n.

Example: σ(6) = 12 = 2(6), so 6 is perfect!

Quick Quiz

What is τ(50)?

Answer: 50 = 2 × 5²

τ(50) = (1 + 1)(2 + 1) = 6


τ(n) is odd ⟺ n is a perfect square

Both appear in distribution of primes and partition theory

Essential for olympiad problems!


For more number theory tricks and problem-solving techniques, visit my YouTube channel "Maths mastery with Dr. Upasana P Taneja"!


Two functions, infinite insights!

Dr. Upasana Pahuja Taneja

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