Sum of divisors function

Definition

The sum of divisors function, denoted as $$ au(n)$$, is a number-theoretic function that calculates the sum of all positive divisors of a given integer $$n$$. This function is not only important for its own properties but also serves as a key example of both additive and multiplicative functions, illustrating how they interact with prime factorization and contribute to the study of number theory.

5 Must Know Facts For Your Next Test

1. The sum of divisors function can be expressed as $$ au(n) = ext{sum}(d | n, d > 0)$$ where the sum is taken over all positive divisors of $$n$$.
2. For prime numbers, the sum of divisors function is given by $$ au(p^k) = 1 + p + p^2 + ... + p^k = \frac{p^{k+1} - 1}{p - 1}$$.
3. The sum of divisors function is a multiplicative function, meaning if two integers are coprime, then the sum of their divisors can be computed as the product of their individual sums of divisors.
4. The sum of divisors function is particularly useful in studying perfect numbers, where a perfect number is defined as one that is equal to the sum of its proper divisors.
5. The behavior of the sum of divisors function can also be analyzed using Dirichlet series, which provides deep insights into its distribution and growth rates.

Review Questions

• How does the sum of divisors function demonstrate properties of additive functions?
  • The sum of divisors function illustrates the characteristics of additive functions by showing that if two integers are coprime, the sum of their divisors can be expressed as the sum of each individual integer's divisors. This means that for coprime integers $$a$$ and $$b$$, we have $$ au(a imes b) = au(a) + au(b)$$. This property allows us to extend our understanding of how divisors interact under multiplication.
• Discuss how the sum of divisors function behaves under prime factorization and its implications for multiplicative functions.
  • The sum of divisors function behaves predictably under prime factorization because it allows us to express it as a product over primes. For an integer $$n$$ with prime factorization $$n = p_1^{k_1} p_2^{k_2} ... p_m^{k_m}$$, the sum of divisors function is given by $$ au(n) = (1 + p_1 + ... + p_1^{k_1})(1 + p_2 + ... + p_2^{k_2})... (1 + p_m + ... + p_m^{k_m})$$. This property shows that it is a multiplicative function and reveals connections between different types of integers based on their prime factors.
• Evaluate the significance of the sum of divisors function in relation to number theory and its broader implications.
  • The significance of the sum of divisors function extends deeply into number theory as it helps identify special classes of numbers such as perfect numbers, which are defined by being equal to the sum of their proper divisors. The connection between this function and concepts like prime factorization highlights its role in understanding the structure and distribution of integers. Additionally, its relationship with multiplicative functions allows mathematicians to delve into advanced topics such as analytic number theory and divisor summatory functions, ultimately impacting fields like cryptography and algorithm design.

Related terms

Additive function: A function $$f$$ is additive if for any two integers $$a$$ and $$b$$ that are coprime, it holds that $$f(a imes b) = f(a) + f(b)$$.

Multiplicative function: A function $$f$$ is multiplicative if for any two coprime integers $$a$$ and $$b$$, it holds that $$f(a imes b) = f(a) imes f(b)$$.

Euler's Totient Function: The Euler's totient function, denoted as $$ ext{φ}(n)$$, counts the positive integers up to a given integer $$n$$ that are coprime to $$n$$.