Additive function#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 of arithmetic functions which are completely additive are:

• The restriction of the logarithmic function to $\backslash N.$
• The multiplicity of a prime factor p in n, that is the largest exponent m for which p^m divides n.
• {{anchor|Integer logarithm}} a₀(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:

::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" {{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

• a₁(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) {{OEIS|A008472}}. 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

From any additive function $f(n)$ it is possible to create a related {{em|multiplicative function}} $g(n),$ which is a function with the property that whenever $a$ and $b$ are coprime then:

$$g(a\; b)\; =\; g(a)\; \backslash times\; g(b).$$

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.

Given an additive function $f$, let its summatory function be defined by $\backslash mathcal\{M\}\_f(x)\; :=\; \backslash sum\_\{n\; \backslash leq\; x\}\; f(n)$. The average of $f$ is given exactly as

$$\backslash mathcal\{M\}\_f(x)\; =\; \backslash sum\_\{p^\{\backslash alpha\}\; \backslash leq\; x\}\; f(p^\{\backslash alpha\})\; \backslash left(\backslash left\backslash lfloor\; \backslash frac\{x\}\{p^\{\backslash alpha\}\}\; \backslash right\backslash rfloor\; -\; \backslash left\backslash lfloor\; \backslash frac\{x\}\{p^\{\backslash alpha+1\}\}\; \backslash right\backslash rfloor\backslash right).$$

The summatory functions over $f$ can be expanded as $\backslash mathcal\{M\}\_f(x)\; =\; x\; E(x)\; +\; O(\backslash sqrt\{x\}\; \backslash cdot\; D(x))$ where

$$\backslash begin\{align\}$$

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}

The average of the function $f^2$ is also expressed by these functions as

$$\backslash mathcal\{M\}\_\{f^2\}(x)\; =\; x\; E^2(x)\; +\; O(x\; D^2(x)).$$

There is always an absolute constant $C\_f\; >\; 0$ such that for all natural numbers $x\; \backslash geq\; 1$,

$$\backslash sum\_\{n\; \backslash leq\; x\}\; |f(n)\; -\; E(x)|^2\; \backslash leq\; C\_f\; \backslash cdot\; x\; D^2(x).$$

Let

$$\backslash nu(x;\; z)\; :=\; \backslash frac\{1\}\{x\}\; \backslash \#\backslash !\backslash left\backslash \{n\; \backslash leq\; x:\; \backslash frac\{f(n)-A(x)\}\{B(x)\}\; \backslash leq\; z\backslash right\backslash \}\backslash !.$$

Suppose that $f$ is an additive function with $-1\; \backslash leq\; f(p^\{\backslash alpha\})\; =\; f(p)\; \backslash leq\; 1$

such that as $x\; \backslash rightarrow\; \backslash infty$,

$$B(x)\; =\; \backslash sum\_\{p\; \backslash leq\; x\}\; f^2(p)\; /\; p\; \backslash rightarrow\; \backslash infty.$$

Then $\backslash nu(x;\; z)\; \backslash sim\; G(z)$ where $G(z)$ is the Gaussian distribution function

$$G(z)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\{z\}\; e^\{-t^2/2\}\; dt.$$

Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed $z\; \backslash in\; \backslash R$ where the relations hold for $x\; \backslash gg\; 1$:

$$\backslash \#\backslash \{n\; \backslash leq\; x:\; \backslash omega(n)\; -\; \backslash log\backslash log\; x\; \backslash leq\; z\; (\backslash log\backslash log\; x)^\{1/2\}\backslash \}\; \backslash sim\; x\; G(z),$$

$$\backslash \#\backslash \{p\; \backslash leq\; x:\; \backslash omega(p+1)\; -\; \backslash log\backslash log\; x\; \backslash leq\; z\; (\backslash log\backslash log\; x)^\{1/2\}\backslash \}\; \backslash sim\; \backslash pi(x)\; G(z).$$

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

{{Reflist}}

{{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).

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Category:Arithmetic functions