Additive function

{{Short description|Function that can be written as a sum over prime factors}} {{About||the algebraic meaning|Additive map}} {{more footnotes|date=February 2013}}

In number theory, an '''{{anchor|definition-additive_function-number_theory}}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'':<ref name="Erdos1939">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. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/ online]</ref> <math display=block>f(a b) = f(a) + f(b).</math> It follows that for any additive function, <math>f(1)=0</math>.

== Completely additive == An additive function ''f''(''n'') is said to be '''completely additive''' if <math>f(a b) = f(a) + f(b)</math> 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.

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 <math>\N.</math> * The '''multiplicity''' of a prime factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''pm'' divides ''n''. * {{anchor|Integer logarithm}} ''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'' {{OEIS|A001414}}. 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" {{OEIS|A001222}}. 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'' {{OEIS|A001221}}. 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'') {{OEIS|A008472}}. 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 <math>f(n)</math> it is possible to create a related {{em|multiplicative function}} <math>g(n),</math> which is a function with the property that whenever <math>a</math> and <math>b</math> are coprime then: <math display=block>g(a b) = g(a) \times g(b).</math> One such example is <math>g(n) = 2^{f(n)}.</math> Likewise if <math>f(n)</math> is completely additive, then <math>g(n) = 2^{f(n)} </math> is completely multiplicative. More generally, we could consider the function <math>g(n) = c^{f(n)} </math>, where <math>c</math> is a nonzero real constant.

== Summatory functions ==

Given an additive function <math>f</math>, let its summatory function be defined by <math display="inline">\mathcal{M}_f(x) := \sum_{n \leq x} f(n)</math>. The average of <math>f</math> is given exactly as <math display=block>\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 display=block>\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 display=block>\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 display=block>\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).</math>

Let <math display=block>\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 display=block>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 display=block>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 \R</math> where the relations hold for <math>x \gg 1</math>: <math display=block>\#\{n \leq x: \omega(n) - \log\log x \leq z (\log\log x)^{1/2}\} \sim x G(z),</math> <math display=block>\#\{p \leq x: \omega(p+1) - \log\log x \leq z (\log\log x)^{1/2}\} \sim \pi(x) G(z).</math>

==References== {{Reflist}}

== Further reading == {{Refbegin}} * 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). {{Refend}}

Category:Arithmetic functions Category:Additive functions