Divisor function

A divisor function assigns a value to each positive integer by counting or summing its positive divisors. The notation σₖ(n) sums the k-th powers of those divisors; τ(n), also written d(n), counts them.

Prime factorization: The expression of an integer as a product of primes, unique up to the order of its factors. Its exponents determine the formulas for counting and summing divisors.
Divisibility: The relation in which an integer a divides an integer b when b is an integer multiple of a. Divisor functions are defined by collecting integers related to n through divisibility.
Perfect number: A positive integer equal to the sum of its positive divisors other than itself. A perfect number satisfies σ(n)=2n.
Euler totient function: The function φ(n) that counts positive integers up to n that are coprime to n. It counts integers defined by coprimality, rather than divisors of n.
Multiplicative function: An arithmetic function f satisfying f(mn)=f(m)f(n) whenever m and n are coprime. Divisor functions are multiplicative, so their values factor across distinct prime powers.
Positive divisor: A positive integer that divides another integer without a remainder. Only positive divisors are included in the standard divisor functions.
Riemann zeta function: The complex function initially defined by ζ(s)=Σₙ₌₁∞ n⁻ˢ where the series converges. The Dirichlet series of σₖ is ζ(s)ζ(s−k) in its region of convergence.
Möbius function: The arithmetic function μ(n) that is zero for integers divisible by a squared prime and otherwise equals (−1) to the number of prime factors. It encodes square-free factorization with signs instead of counting or summing divisors.
Divisor-counting function: The function τ(n), also written d(n), that counts the positive divisors of n. It is the divisor function that counts rather than sums divisors.
Arithmetic function: A function defined on the positive integers, usually with values in the real or complex numbers. A divisor function is a particular arithmetic function.

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