In number theory, $\tau(n)$ and $\sigma(n)$ are two important arithmetic functions related to the divisors of a positive integer $n$. Here are their definitions:

1. $\tau(n)$ (Divisor Function or Divisor Count Function)

$\tau(n)$, also denoted as $d(n)$, is the number of divisors of $n$. For example, if $n = 6$, the divisors of 6 are $1, 2, 3, 6$, so: $\tau(6) = 4$ More formally: $\tau(n) = \sum_{d \mid n} 1$ where $d \mid n$ means that $d$ is a divisor of $n$.

2. $\sigma(n)$ (Divisor Sum Function)

$\sigma(n)$ is the sum of all positive divisors of $n$. Using the same example $n = 6$, the divisors of 6 are $1, 2, 3, 6$, so: $\sigma(6) = 1 + 2 + 3 + 6 = 12$ More formally: $\sigma(n) = \sum_{d \mid n} d$ This function sums the divisors of $n$ instead of just counting them.

Example Calculation

For $n = 12$:

• The divisors of 12 are $1, 2, 3, 4, 6, 12$.
• Therefore:
  • $\tau(12) = 6$ (because 12 has 6 divisors).
  • $\sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28$.

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Would you like more details on their properties or applications?

5 Related Questions:

1. How do you compute $\tau(n)$ and $\sigma(n)$ for large numbers efficiently?
2. What is the relationship between $\tau(n)$ and the prime factorization of $n$?
3. Can $\sigma(n)$ be used to classify numbers as perfect, deficient, or abundant?
4. How does $\tau(n)$ behave for prime numbers versus composite numbers?
5. What are the generalizations of $\sigma(n)$ in modular arithmetic?

Tip: The functions $\tau(n)$ and $\sigma(n)$ are examples of multiplicative functions, meaning that if $m$ and $n$ are coprime, then $\tau(mn) = \tau(m) \cdot \tau(n)$ and similarly for $\sigma(n)$.