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Additive function

Function that can be written as a sum over prime factors

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated May 11, 2026

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(ab)=f(a)+f(b). It follows that for any additive function, f(1)=0.

01Completely additive

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.

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

02Examples

Examples of arithmetic functions which are completely additive are:

  • The restriction of the logarithmic function to \mathbb {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 (sequence A001414 in the OEIS). 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" (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(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(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) (sequence A008472 in the OEIS). 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

03Multiplicative functions

From any additive function f(n) it is possible to create a related multiplicative function g(n), which is a function with the property that whenever a and b are coprime then: g(ab)=g(a)\times g(b). One such example is g(n)=2^{f(n)}. Likewise if f(n) is completely additive, then g(n)=2^{f(n)} is completely multiplicative. More generally, we could consider the function g(n)=c^{f(n)}, where c is a nonzero real constant.

04Summatory functions

Given an additive function f, let its summatory function be defined by {\textstyle {\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)=xE(x)+O({\sqrt {x}}\cdot D(x)) where {\begin{aligned}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{aligned}}

The average of the function f^{2} is also expressed by these functions as {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{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 xD^{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 \mathbb {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 xG(z), \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).

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Sources and credits

This article is adapted from the Wikipedia article “Additive function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Sigma-additive set function

In mathematics, an additive set function is a function μ \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, μ = μ ( A ) + μ ( B ) . {\textstyle \mu (A\cup B)=\mu (A)+\mu (B).} If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets.

Prime omega function

In number theory, the prime omega functions ω {\displaystyle \omega (n)} and Ω ( n ) {\displaystyle \Omega (n)} count the number of prime factors of a natural number n {\displaystyle n} . The number of distinct prime factors is assigned to ω ( n ) {\displaystyle \omega (n)} (little omega), while Ω ( n ) {\displaystyle \Omega (n)} (big omega) counts the total number of prime factors with multiplicity (see arithmetic function).

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that f = 1 {\displaystyle f(1)=1} and f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} whenever a {\displaystyle a} and b {\displaystyle b} are coprime. An arithmetic function is said to be completely multiplicative (or totally multiplicative) if f ( 1 ) = 1 {\displaystyle f(1)=1} and f ( a b ) = f ( a ) f ( b ) {\displaystyle f(ab)=f(a)f(b)} holds for all positive integers a {\displaystyle a} and b {\displaystyle b} , even when they are not coprime.