Additive function

In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: $$f(a b) = f(a) + f(b).$$

Completely additive
An additive function f(n) is said to be completely additive if $$f(a b) = f(a) + f(b)$$ holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0.

Every completely additive function is additive, but not vice versa.

Examples
Examples of arithmetic functions which are completely additive are:


 * The restriction of the logarithmic function to $$\N.$$
 * The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.
 * a0(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n . For example:


 * a0(4) = 2 + 2 = 4
 * a0(20) = a0(22 · 5) = 2 + 2 + 5 = 9
 * a0(27) = 3 + 3 + 3 = 9
 * a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
 * a0(2000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
 * a0(2003) = 2003
 * a0(54,032,858,972,279) = 1240658
 * a0(54,032,858,972,302) = 1780417
 * a0(20,802,650,704,327,415) = 1240681


 * The function Ω(n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" . For example;


 * Ω(1) = 0, since 1 has no prime factors
 * Ω(4) = 2
 * Ω(16) = Ω(2·2·2·2) = 4
 * Ω(20) = Ω(2·2·5) = 3
 * Ω(27) = Ω(3·3·3) = 3
 * Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
 * Ω(2000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
 * Ω(2001) = 3
 * Ω(2002) = 4
 * Ω(2003) = 1
 * Ω(54,032,858,972,279) = Ω(11 ⋅ 19932 ⋅ 1236661) = 4  ;
 * Ω(54,032,858,972,302) = Ω(2 ⋅ 72 ⋅ 149 ⋅ 2081 ⋅ 1778171) = 6
 * Ω(20,802,650,704,327,415) = Ω(5 ⋅ 7 ⋅ 112 ⋅ 19932 ⋅ 1236661) = 7.

Examples of arithmetic functions which are additive but not completely additive are:


 * ω(n), defined as the total number of distinct prime factors of n . For example:


 * ω(4) = 1
 * ω(16) = ω(24) = 1
 * ω(20) = ω(22 · 5) = 2
 * ω(27) = ω(33) = 1
 * ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
 * ω(2000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2
 * ω(2001) = 3
 * ω(2002) = 4
 * ω(2003) = 1
 * ω(54,032,858,972,279) = 3
 * ω(54,032,858,972,302) = 5
 * ω(20,802,650,704,327,415) = 5


 * a1(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) . For example:


 * a1(1) = 0
 * a1(4) = 2
 * a1(20) = 2 + 5 = 7
 * a1(27) = 3
 * a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
 * a1(2000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
 * a1(2001) = 55
 * a1(2002) = 33
 * a1(2003) = 2003
 * a1(54,032,858,972,279) = 1238665
 * a1(54,032,858,972,302) = 1780410
 * a1(20,802,650,704,327,415) = 1238677

Multiplicative functions
From any additive function $$f(n)$$ it is possible to create a related $$g(n),$$ which is a function with the property that whenever $$a$$ and $$b$$ are coprime then: $$g(a b) = g(a) \times g(b).$$ One such example is $$g(n) = 2^{f(n)}.$$

Summatory functions
Given an additive function $$f$$, let its summatory function be defined by $\mathcal{M}_f(x) := \sum_{n \leq x} f(n)$. The average of $$f$$ is given exactly as $$\mathcal{M}_f(x) = \sum_{p^{\alpha} \leq x} f(p^{\alpha}) \left(\left\lfloor \frac{x}{p^{\alpha}} \right\rfloor - \left\lfloor \frac{x}{p^{\alpha+1}} \right\rfloor\right).$$

The summatory functions over $$f$$ can be expanded as $$\mathcal{M}_f(x) = x E(x) + O(\sqrt{x} \cdot D(x))$$ where $$\begin{align} E(x) & = \sum_{p^{\alpha} \leq x} f(p^{\alpha}) p^{-\alpha} (1-p^{-1}) \\ D^2(x) & = \sum_{p^{\alpha} \leq x} |f(p^{\alpha})|^2 p^{-\alpha}. \end{align}$$

The average of the function $$f^2$$ is also expressed by these functions as $$\mathcal{M}_{f^2}(x) = x E^2(x) + O(x D^2(x)).$$

There is always an absolute constant $$C_f > 0$$ such that for all natural numbers $$x \geq 1$$, $$\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).$$

Let $$\nu(x; z) := \frac{1}{x} \#\!\left\{n \leq x: \frac{f(n)-A(x)}{B(x)} \leq z\right\}\!.$$

Suppose that $$f$$ is an additive function with $$-1 \leq f(p^{\alpha}) = f(p) \leq 1$$ such that as $$x \rightarrow \infty$$, $$B(x) = \sum_{p \leq x} f^2(p) / p \rightarrow \infty.$$

Then $$\nu(x; z) \sim G(z)$$ where $$G(z)$$ is the Gaussian distribution function $$G(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2} dt.$$

Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed $$z \in \R$$ where the relations hold for $$x \gg 1$$: $$\#\{n \leq x: \omega(n) - \log\log x \leq z (\log\log x)^{1/2}\} \sim x G(z),$$ $$\#\{p \leq x: \omega(p+1) - \log\log x \leq z (\log\log x)^{1/2}\} \sim \pi(x) G(z).$$