product over all the primes, where all but a finite number have a zero exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then νp(n) = ai, otherwise it is zero. Then In terms of the above the prime omega functions ω and Ω are defined by To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of n and the corresponding pi, ai, ω, and Ω. Multiplicative functions σk(n), τ(n), d(n) – divisor sums σk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number. σ1(n), the sum of the (positive) divisors of n, is usually denoted by σ(n). Since a positive number to the zero power is one, σ0(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or τ(n) (for the German Teiler = divisors). Setting k = 0 in the second product gives φ(n) – Euler totient function φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n. Jk(n) – Jordan totient function Jk(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. It is a generalization of Euler's totient, . μ(n) – Möbius function μ(n), the Möbius function, is important because of the Möbius inversion formula. See Dirichlet convolution, below. This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.) τ(n) – Ramanujan tau function τ(n), the Ramanujan tau function, is defined by its generating function identity: Although it is hard to say exactly what "arithmetical property of n" it "expresses", (τ(n) is (2π)−12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function) it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σk(n) and rk(n) functions (because these are also coefficients in the expansion of modular forms). cq(n) – Ramanujan's sum cq(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity: Even though it is defined as a sum of complex numbers (irrational for most values