I thought I would post this because it may be helpful for us this semester.. This is straight from wikipedia…

Examples of multiplicative functions include many functions of importance in number theory, such as:

* φ(n): Euler's totient function φ, counting the positive integers coprime to (but not bigger than) n

* μ(n): the Möbius function, related to the number of prime factors of square-free numbers

* gcd(n,k): the greatest common divisor of n and k, where k is a fixed integer.

* d(n): the number of positive divisors of n,

* σ(n): the sum of all the positive divisors of n,

* σk(n): the divisor function, which is the sum of the k-th powers of all the positive divisors of n (where k may be any complex number). In special cases we have

o σ0(n) = d(n) and

o σ1(n) = σ(n),

* a(n): the number of non-isomorphic abelian groups of order n.

* 1(n): the constant function, defined by 1(n) = 1 (completely multiplicative)

* 1C(n) the indicator function of the set C of squares (or cubes, or fourth powers, etc.)

* Id(n): identity function, defined by Id(n) = n (completely multiplicative)

* Idk(n): the power functions, defined by Idk(n) = nk for any natural (or even complex) number k (completely multiplicative). As special cases we have

o Id0(n) = 1(n) and

o Id1(n) = Id(n),

* ε(n): the function defined by ε(n) = 1 if n = 1 and = 0 if n > 1, sometimes called multiplication unit for Dirichlet convolution or simply the unit function; sometimes written as u(n), not to be confused with μ(n) (completely multiplicative).

* (n/p), the Legendre symbol, where p is a fixed prime number (completely multiplicative).

* λ(n): the Liouville function, related to the number of prime factors dividing n (completely multiplicative).

* γ(n), defined by γ(n)=(-1)ω(n), where the additive function ω(n) is the number of distinct primes dividing n.

* All Dirichlet characters are completely multiplicative functions.