# Additive function

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In number theory, an **additive function** is an arithmetic function *f*(*n*) of the positive integer *n* such that whenever *a* and *b* are coprime, the function of the product is the sum of the functions:^{[1]}

*f*(*ab*) =*f*(*a*) +*f*(*b*).

## Completely additiveEdit

An additive function *f*(*n*) is said to be **completely additive** if *f*(*ab*) = *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.

## ExamplesEdit

Example 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*p*divides^{m}*n*. *a*_{0}(*n*) - the sum of primes dividing*n*counting multiplicity, sometimes called sopfr(*n*), the potency of*n*or the integer logarithm of*n*(sequence A001414 in the OEIS). For example:

*a*_{0}(4) = 2 + 2 = 4*a*_{0}(20) =*a*_{0}(2^{2}· 5) = 2 + 2+ 5 = 9*a*_{0}(27) = 3 + 3 + 3 = 9*a*_{0}(144) =*a*_{0}(2^{4}· 3^{2}) =*a*_{0}(2^{4}) +*a*_{0}(3^{2}) = 8 + 6 = 14*a*_{0}(2,000) =*a*_{0}(2^{4}· 5^{3}) =*a*_{0}(2^{4}) +*a*_{0}(5^{3}) = 8 + 15 = 23*a*_{0}(2,003) = 2003*a*_{0}(54,032,858,972,279) = 1240658*a*_{0}(54,032,858,972,302) = 1780417*a*_{0}(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" (sequence A001222 in the OEIS). 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(2
^{4}· 3^{2}) = Ω(2^{4}) + Ω(3^{2}) = 4 + 2 = 6 - Ω(2,000) = Ω(2
^{4}· 5^{3}) = Ω(2^{4}) + Ω(5^{3}) = 4 + 3 = 7 - Ω(2,001) = 3
- Ω(2,002) = 4
- Ω(2,003) = 1
- Ω(54,032,858,972,279) = 3
- Ω(54,032,858,972,302) = 6
- Ω(20,802,650,704,327,415) = 7

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

- ω(
*n*), defined as the total number of*different*prime factors of*n*(sequence A001221 in the OEIS). For example:

- ω(4) = 1
- ω(16) = ω(2
^{4}) = 1 - ω(20) = ω(2
^{2}· 5) = 2 - ω(27) = ω(3
^{3}) = 1 - ω(144) = ω(2
^{4}· 3^{2}) = ω(2^{4}) + ω(3^{2}) = 1 + 1 = 2 - ω(2,000) = ω(2
^{4}· 5^{3}) = ω(2^{4}) + ω(5^{3}) = 1 + 1 = 2 - ω(2,001) = 3
- ω(2,002) = 4
- ω(2,003) = 1
- ω(54,032,858,972,279) = 3
- ω(54,032,858,972,302) = 5
- ω(20,802,650,704,327,415) = 5

*a*_{1}(*n*) - the sum of the distinct primes dividing*n*, sometimes called sopf(*n*) (sequence A008472 in the OEIS). For example:

*a*_{1}(1) = 0*a*_{1}(4) = 2*a*_{1}(20) = 2 + 5 = 7*a*_{1}(27) = 3*a*_{1}(144) =*a*_{1}(2^{4}· 3^{2}) =*a*_{1}(2^{4}) +*a*_{1}(3^{2}) = 2 + 3 = 5*a*_{1}(2,000) =*a*_{1}(2^{4}· 5^{3}) =*a*_{1}(2^{4}) +*a*_{1}(5^{3}) = 2 + 5 = 7*a*_{1}(2,001) = 55*a*_{1}(2,002) = 33*a*_{1}(2,003) = 2003*a*_{1}(54,032,858,972,279) = 1238665*a*_{1}(54,032,858,972,302) = 1780410*a*_{1}(20,802,650,704,327,415) = 1238677

## Multiplicative functionsEdit

From any additive function *f*(*n*) it is easy to create a related multiplicative function *g*(*n*) i.e. with the property that whenever *a* and *b* are coprime we have:

*g*(*ab*) =*g*(*a*) ×*g*(*b*).

One such example is *g*(*n*) = 2^{f(n)}.

## Summatory functionsEdit

Given an additive function <math>f</math>, let its summatory function be defined by <math>\mathcal{M}_f(x) := \sum_{n \leq x} f(n)</math>. The average of <math>f</math> is given exactly as

- <math> \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). </math>

The summatory functions over <math>f</math> can be expanded as <math>\mathcal{M}_f(x) = x E(x) + O(\sqrt{x} \cdot D(x))</math> where

- <math> \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} </math>

The average of the function <math>f^2</math> is also expressed by these functions as

- <math>\mathcal{M}_{f^2}(x) = x E^2(x) + O(x D^2(x)).</math>

There is always an absolute constant <math>C_f > 0</math> such that for all natural numbers <math>x \geq 1</math>,

- <math>\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).</math>

Let

- <math> \nu(x; z) := \frac{1}{x} \#\left\{n \leq x: \frac{f(n)-A(x)}{B(x)} \leq z\right\}.</math>

Suppose that <math>f</math> is an additive function with <math>-1 \leq f(p^{\alpha}) = f(p) \leq 1</math> such that as <math>x \rightarrow \infty</math>,

- <math>B(x) = \sum_{p \leq x} f^2(p) / p \rightarrow \infty. </math>

Then <math>\nu(x; z) \sim G(z)</math> where <math>G(z)</math> is the Gaussian distribution function

- <math>G(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2} dt.</math>

Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed <math>z \in \mathbb{R}</math> where the relations hold for <math>x \gg 1</math>:

- <math>\#\{n \leq x: \omega(n) - \log\log x \leq z (\log\log x)^{1/2}\} \sim x G(z), </math>
- <math>\#\{p \leq x: \omega(p+1) - \log\log x \leq z (\log\log x)^{1/2}\} \sim \pi(x) G(z). </math>

## See alsoEdit

## ReferencesEdit

## Further readingEdit

- Janko Bračič,
*Kolobar aritmetičnih funkcij*(*Ring of arithmetical functions*), (Obzornik mat, fiz.**49**(2002) 4, pp. 97–108) (MSC (2000) 11A25) - Iwaniec and Kowalski,
*Analytic number theory*, AMS (2004).