Additive function

In number theory, an Template:Anchoradditive 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: $$f(ab)=f(a)+f(b).$$

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 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 $\mathbb{N}.$
• The multiplicity of a prime factor p in n, that is the largest exponent m for which p^m divides n.
• Template:Anchor a₀(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n Template: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" Template: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(n), defined as the total number of distinct prime factors of n Template:OEIS. 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

• a₁(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) Template: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

Multiplicative functions

From any additive function $f(n)$ it is possible to create a related Template:Em $g(n),$ 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.

Summatory functions

Given an additive function $f$, let its summatory function be defined by ${ℳ}_f(x):=\sum _{n\le x}f(n)$. The average of $f$ is given exactly as $${ℳ}_f(x)=\sum _{p^{\alpha }\le x}f(p^{\alpha })(\lfloor \frac{x}{p^{\alpha }}\rfloor -\lfloor \frac{x}{p^{\alpha +1}}\rfloor ).$$

The summatory functions over $f$ can be expanded as ${ℳ}_f(x)=xE(x)+O(\sqrt{x}\cdot D(x))$ where $$\begin{array}{cc}E(x)& =\sum _{p^{\alpha }\le x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\ D^2(x)& =\sum _{p^{\alpha }\le x}|f(p^{\alpha }){|}^2{p}^{-\alpha }.\end{array}$$

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

There is always an absolute constant $C_f>0$ such that for all natural numbers $x\ge 1$, $$\sum _{n\le x}|f(n)-E(x){|}^2\le C_f\cdot x{D}^2(x).$$

Let $$\nu (x;z):=\frac{1}{x}\mathrm{\#}\{n\le x:\frac{f(n)-A(x)}{B(x)}\le z\}.$$

Suppose that $f$ is an additive function with $-1\le f(p^{\alpha })=f(p)\le 1$ such that as $x\to \mathrm{\infty }$, $$B(x)=\sum _{p\le x}{f}^2(p)/p\to \mathrm{\infty }.$$

Then $\nu (x;z)\sim G(z)$ where $G(z)$ is the Gaussian distribution function $$G(z)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{z}{\int }}{e}^{-t^2/2}dt.$$

Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed $z\in ℝ$ where the relations hold for $x\gg 1$: $$\mathrm{\#}\{n\le x:\omega (n)-\log \log x\le z(\log \log x{)}^{1/2}\}\sim xG(z),$$ $$\mathrm{\#}\{p\le x:\omega (p+1)-\log \log x\le z(\log \log x{)}^{1/2}\}\sim \pi (x)G(z).$$

See also

• Sigma additivity
• Prime omega function
• Multiplicative function
• Arithmetic function

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).