Arithmetic function

An arithmetic function maps positive integers to values, usually integers or complex numbers. Its properties are studied through how those values vary across the integers.

Divisibility: The relation in which one integer is an exact multiple of another. Many arithmetic functions depend on which integers divide their input.
Dirichlet series: A series of the form sum a(n)n^(-s), used to study sequences through complex analysis. An arithmetic function can be represented by its values as coefficients of a Dirichlet series.
Divisor function: The function σ_k(n) that sums the k-th powers of the positive divisors of n. It packages the number and sizes of an integer’s divisors into one value.
Completely multiplicative function: An arithmetic function satisfying f(mn)=f(m)f(n) for all positive integers m and n. Unlike ordinary multiplicativity, this condition also constrains products of non-coprime inputs.
Prime number: A natural number greater than one whose only positive divisors are one and itself. Prime inputs often determine an arithmetic function’s behavior on all integers.
Möbius inversion formula: A formula that recovers a function from its sums over divisors using the Möbius function. It reverses divisor-sum relations between arithmetic functions.
Euler's totient function: The function φ(n) counting positive integers up to n that are coprime to n. It measures how many residue classes modulo n are invertible.
Additive function: An arithmetic function satisfying f(mn)=f(m)+f(n) whenever m and n are coprime. Its coprime-product rule uses addition rather than the multiplication defining multiplicative functions.
Fundamental theorem of arithmetic: The theorem that every integer greater than one has a unique factorization into primes, up to ordering. Multiplicative functions are reconstructed from values at prime powers using unique factorization.
Euler product: A product indexed by primes that represents certain Dirichlet series when their coefficients are multiplicative. Multiplicativity lets a function’s Dirichlet series factor into contributions from prime powers.

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