Additive function

For the algebraic meaning, see Additive map.

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:

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

Completely 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 co-prime. 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

Example 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 $m\in \mathbb {N}$ for which $p^{m}|n$ is true.

• $a_{0}(n)$ - the sum of primes dividing $n$, counting multiplicity, sometimes called ${\text{sopfr}}(n)$, the potency of $n$ or the integer logarithm of $n$ (sequence A001414 in the OEIS). For example:

$a_{0}(4)=2+2=4$

$a_{0}(20)=a_{0}(2^{2}\cdot 5)=2+2+5=9$

$a_{0}(27)=3+3+3=9$

$a_{0}(144)=a_{0}(2^{4}\cdot 3^{2})=a_{0}(2^{4})+a_{0}(3^{2})=8+6=14$

$a_{0}(2,000)=a_{0}(2^{4}\cdot 5^{3})=a_{0}(2^{4})+a_{0}(5^{3})=8+15=23$

$a_{0}(2,003)=2,003$

$a_{0}(54,032,858,972,279)=1,240,658$

$a_{0}(54,032,858,972,302)=1,780,417$

$a_{0}(20,802,650,704,327,415)=1,240,681$

• The omega function $\Omega (n)$, defined as the total number of prime factors of $n$, counting multiplicity. It is sometimes called the "Big Omega function" (sequence A001222 in the OEIS). For example;

$\Omega (1)=0$, since 1 has no prime factors

$\Omega (4)=2$

$\Omega (16)=\Omega (2\cdot 2\cdot 2\cdot 2)=4$

$\Omega (20)=\Omega (2\cdot 2\cdot 5)=3$

$\Omega (27)=\Omega (3\cdot 3\cdot 3)=3$

$\Omega (144)=\Omega (2^{4}\cdot 3^{2})=\Omega (2^{4})+\Omega (3^{2})=4+2=6$

$\Omega (2,000)=\Omega (2^{4}\cdot 5^{3})=\Omega (2^{4})+\Omega (5^{3})=4+3=7$

$\Omega (2,001)=3$

$\Omega (2,002)=4$

$\Omega (2,003)=1$

$\Omega (54,032,858,972,279)=3$

$\Omega (54,032858972,302)=6$

$\Omega (20,802,650,704,327,415)=7$

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

• The other omega function, $\omega (n)$, defined as the number of distinct prime factors of $n$ (sequence A001221 in the OEIS). For example:

$\omega (4)=1$

$\omega (16)=\omega (2^{4})=1$

$\omega (20)=\omega (2^{2}\cdot 5)=2$

$\omega (27)=\omega (3^{3})=1$

$\omega (144)=\omega (2^{4}\cdot 3^{2})=\omega (2^{4})+\omega (3^{2})=1+1=2$

$\omega (2,000)=\omega (2^{4}\cdot 5^{3})=\omega (2^{4})+\omega (5^{3})=1+1=2$

$\omega (2,001)=3$

$\omega (2,002)=4$

$\omega (2,003)=1$

$\omega (54,032,858,972,279)=3$

$\omega (54,032,858,972,302)=5$

$\omega (20,802,650,704,327,415)=5$

• $a_{1}(n)$ - the sum of the distinct primes dividing $n$, sometimes called ${\text{sopf}}(n)$ (sequence A008472 in the OEIS). For example:

$a_{1}(1)=0$

$a_{1}(4)=2$

$a_{1}(20)=2+5=7$

$a_{1}(27)=3$

$a_{1}(144)=a_{1}(2^{4}\cdot 3^{2})=a_{1}(2^{4})+a_{1}(3^{2})=2+3=5$

$a_{1}(2,000)=a_{1}(2^{4}\cdot 5^{3})=a_{1}(2^{4})+a_{1}(5^{3})=2+5=7$

$a_{1}(2,001)=55$

$a_{1}(2,002)=33$

$a_{1}(2,003)=2003$

$a_{1}(54,032,858,972,279)=1,238,665$

$a_{1}(54,032,858,972,302)=1,780,410$

$a_{1}(20,802,650,704,327,415)=1,238,677$

Multiplicative functions

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:

$g(ab)=g(a)\cdot g(b)$.

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

See also

• Sigma additivity

Further reading

• Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)