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'':Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207
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$f(a b) = f(a) + f(b).$

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 ''p^m'' divides ''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'' . For example:

::''a''₀(4) = 2 + 2 = 4
::''a''₀(20) = ''a''₀(2² · 5) = 2 + 2 + 5 = 9
::''a''₀(27) = 3 + 3 + 3 = 9
::''a''₀(144) = ''a''₀(2⁴ · 3²) = ''a''₀(2⁴) + ''a''₀(3²) = 8 + 6 = 14
::''a''₀(2000) = ''a''₀(2⁴ · 5³) = ''a''₀(2⁴) + ''a''₀(5³) = 8 + 15 = 23
::''a''₀(2003) = 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" . 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

* ''a''₁(''n'') – the sum of the distinct primes dividing ''n'', sometimes called sopf(''n'') . 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(2⁴ · 3²) = ''a''₁(2⁴) + ''a''₁(3²) = 2 + 3 = 5
::''a''₁(2000) = ''a''₁(2⁴ · 5³) = ''a''₁(2⁴) + ''a''₁(5³) = 2 + 5 = 7
::''a''₁(2001) = 55
::''a''₁(2002) = 33
::''a''₁(2003) = 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 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 $g(n) = 2^.$ Likewise if $f(n)$ is completely additive, then $g(n) = 2^$ is completely multiplicative. More generally, we could consider the function $g(n) = c^$, where $c$ is a nonzero real constant.

Summatory functions

Given an additive function $f$, let its summatory function be defined by $\mathcal_f(x) := \sum_ f(n)$. The average of $f$ is given exactly as
$\mathcal_f(x) = \sum_ f(p^) \left(\left\lfloor \frac \right\rfloor - \left\lfloor \frac \right\rfloor\right).$

The summatory functions over $f$ can be expanded as $\mathcal_f(x) = x E(x) + O(\sqrt \cdot D(x))$ where
$\begin E(x) & = \sum_ f(p^) p^ (1-p^) \\ D^2(x) & = \sum_ , f(p^), ^2 p^. \end$

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

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

Let
$\nu(x; z) := \frac \#\!\left\\!.$

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

Then $\nu(x; z) \sim G(z)$ where $G(z)$ is the Gaussian distribution function
$G(z) = \frac \int_^ e^ dt.$

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 $x \gg 1$:
$\#\ \sim x G(z),$
$\#\ \sim \pi(x) G(z).$

See also

* Sigma additivity
* Prime omega function
* Multiplicative function
* Arithmetic function

References

Further reading

* 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).

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Arithmetic functions