Additive Functions

Subjects: Elementary Number Theory, Analytic Number Theory
Links: Multiplicative Functions

Def: A arithmetic function $f$ is called additive if

f (m n) = f (m) + f (n)

for $gcd (m, n) = 1$

Def: A arithmetic function $f$ is called completely additive if for any $m, n$

f (m n) = f (m) + f (n)

Th: If we have an additive function $f$ , then we can get a multiplicative function by

g (n) = b^{f (n)}

for a fixed base $b$ , and if the function $f$ is completely additive then $g$ will be completely multiplicative.

Prime Omega functions

If $n = p_{1}^{k_{1}} p_{2}^{k_{2}} \dots p_{r}^{k_{r}}$ is the prime factorization of $n$ , then we the define $ω (n)$ to be the number of distinct prime factors,

ω (n) = \sum_{p ∣ n} 1

and, $Ω (n)$ counts the total number of prime factors of $n$

Ω (n) = \sum_{p^{α} ∣ n} 1 = \sum_{p^{α} ∥ n} α = \sum_{i = 1}^{r} k_{i}

Which is completely additive, and has the following properties

ω (n) \leq Ω (n)

Liouville function

From these two prime counting functions we can get that another important functions called the Liouville function defined as

λ (n) = (- 1)^{Ω (n)}

it is completely multiplicative.

If we look at the

\sum_{d ∣ n} λ (d) = {\begin{cases} 1 & if n is a perfect square \\ 0 & otherwise \end{cases}

which is the square number characteristic function.