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In `F33f`_`[number theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Number_theory]`_`f, an `!additive function`! is an `F33f`_`[arithmetic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_function]`_`f `*f`*(`*n`*) of the positive `F33f`_`[integer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Integer]`_`f variable `*n`* such that whenever `*a`* and `*b`* are `F33f`_`[coprime`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Coprime]`_`f, the function applied to the product `*ab`* is the sum of the values of the function applied to `*a`* and `*b`*:`:cite-ref-erdos1939-1-0[`F5bf`_`[1`#cite-note-erdos1939-1]`_`f] f ( a b ) = f ( a ) + f ( b ) . {\\displaystyle f(ab)=f(a)+f(b).}

>>Contents

• `F0af`_`[Completely additive`#completely-additive]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[Multiplicative functions`#multiplicative-functions]`_`f

• `F0af`_`[Summatory functions`#summatory-functions]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Further reading`#further-reading]`_`f

-─

>>Completely additive

An additive function `*f`*(`*n`*) is said to be `!completely additive`! if f ( a b ) = f ( a ) + f ( b ) {\\displaystyle 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 `F33f`_`[totally multiplicative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Totally_multiplicative]`_`f 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 `F33f`_`[logarithmic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f to N . {\\displaystyle \\mathbb {N} .}

• The `!multiplicity`! of a `F33f`_`[prime`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_number]`_`f factor `*p`* in `*n`*, that is the largest exponent `*m`* for which `*pm`* `F33f`_`[divides`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Divisor]`_`f `*n`*.

• `*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`* (sequence A001414 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f). 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 `F33f`_`[prime factors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_factor]`_`f of `*n`*, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence A001222 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f). 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 `F33f`_`[prime factors`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_factor]`_`f of `*n`* (sequence A001221 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f). 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`*) (sequence A008472 in the `F33f`_`[OEIS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=On-Line_Encyclopedia_of_Integer_Sequences]`_`f). 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 f ( n ) {\\displaystyle f(n)} it is possible to create a related `*`F33f`_`[multiplicative function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multiplicative_function]`_`f`* g ( n ) , {\\displaystyle g(n),} which is a function with the property that whenever a {\\displaystyle a} and b {\\displaystyle b} are coprime then: g ( a b ) = g ( a ) × × g ( b ) . {\\displaystyle g(ab)=g(a)\\times g(b).} One such example is g ( n ) = 2 f ( n ) . {\\displaystyle g(n)=2^{f(n)}.} Likewise if f ( n ) {\\displaystyle f(n)} is completely additive, then g ( n ) = 2 f ( n ) {\\displaystyle g(n)=2^{f(n)}} is completely multiplicative. More generally, we could consider the function g ( n ) = c f ( n ) {\\displaystyle g(n)=c^{f(n)}} , where c {\\displaystyle c} is a nonzero real constant.

>>Summatory functions

Given an additive function f {\\displaystyle f} , let its summatory function be defined by M f ( x ) := ∑ ∑ n ≤ ≤ x f ( n ) {\\textstyle {\\mathcal {M}}_{f}(x):=\\sum _{n\\leq x}f(n)} . The average of f {\\displaystyle f} is given exactly as M f ( x ) = ∑ ∑ p α α ≤ ≤ x f ( p α α ) ( ⌊ x p α α ⌋ − − ⌊ x p α α + 1 ⌋ ) . {\\displaystyle {\\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 {\\displaystyle f} can be expanded as M f ( x ) = x E ( x ) + O ( x ⋅ ⋅ D ( x ) ) {\\displaystyle {\\mathcal {M}}_{f}(x)=xE(x)+O({\\sqrt {x}}\\cdot D(x))} where E ( x ) = ∑ ∑ p α α ≤ ≤ x f ( p α α ) p − − α α ( 1 − − p − − 1 ) D 2 ( x ) = ∑ ∑ p α α ≤ ≤ x | f ( p α α ) | 2 p − − α α . {\\displaystyle {\\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 {\\displaystyle f^{2}} is also expressed by these functions as M f 2 ( x ) = x E 2 ( x ) + O ( x D 2 ( x ) ) . {\\displaystyle {\\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).}

There is always an absolute constant C f > 0 {\\displaystyle C_{f}>0} such that for all `F33f`_`[natural numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Natural_number]`_`f x ≥ ≥ 1 {\\displaystyle x\\geq 1} , ∑ ∑ n ≤ ≤ x | f ( n ) − − E ( x ) | 2 ≤ ≤ C f ⋅ ⋅ x D 2 ( x ) . {\\displaystyle \\sum _{n\\leq x}|f(n)-E(x)|^{2}\\leq C_{f}\\cdot xD^{2}(x).}

Let ν ν ( x ; z ) := 1 x # # { n ≤ ≤ x : f ( n ) − − A ( x ) B ( x ) ≤ ≤ z } . {\\displaystyle \\nu (x;z):={\\frac {1}{x}}\\#\\!\\left\\{n\\leq x:{\\frac {f(n)-A(x)}{B(x)}}\\leq z\\right\\}\\!.}

Suppose that f {\\displaystyle f} is an additive function with − − 1 ≤ ≤ f ( p α α ) = f ( p ) ≤ ≤ 1 {\\displaystyle -1\\leq f(p^{\\alpha })=f(p)\\leq 1} such that as x → → ∞ ∞ {\\displaystyle x\\rightarrow \\infty } , B ( x ) = ∑ ∑ p ≤ ≤ x f 2 ( p ) / p → → ∞ ∞ . {\\displaystyle B(x)=\\sum _{p\\leq x}f^{2}(p)/p\\rightarrow \\infty .}

Then ν ν ( x ; z ) ∼ ∼ G ( z ) {\\displaystyle \\nu (x;z)\\sim G(z)} where G ( z ) {\\displaystyle G(z)} is the `F33f`_`[Gaussian distribution function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f G ( z ) = 1 2 π π ∫ ∫ − − ∞ ∞ z e − − t 2 / 2 d t . {\\displaystyle G(z)={\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{z}e^{-t^{2}/2}dt.}

Examples of this result related to the `F33f`_`[prime omega function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_omega_function]`_`f and the numbers of prime divisors of shifted primes include the following for fixed z ∈ ∈ R {\\displaystyle z\\in \\mathbb {R} } where the relations hold for x ≫ ≫ 1 {\\displaystyle x\\gg 1} : # # { n ≤ ≤ x : ω ω ( n ) − − log ⁡ ⁡ log ⁡ ⁡ x ≤ ≤ z ( log ⁡ ⁡ log ⁡ ⁡ x ) 1 / 2 } ∼ ∼ x G ( z ) , {\\displaystyle \\#\\{n\\leq x:\\omega (n)-\\log \\log x\\leq z(\\log \\log x)^{1/2}\\}\\sim xG(z),} # # { p ≤ ≤ x : ω ω ( p + 1 ) − − log ⁡ ⁡ log ⁡ ⁡ x ≤ ≤ z ( log ⁡ ⁡ log ⁡ ⁡ x ) 1 / 2 } ∼ ∼ π π ( x ) G ( z ) . {\\displaystyle \\#\\{p\\leq x:\\omega (p+1)-\\log \\log x\\leq z(\\log \\log x)^{1/2}\\}\\sim \\pi (x)G(z).}

>>See also

• `F33f`_`[Sigma additivity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sigma_additivity]`_`f

• `F33f`_`[Prime omega function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prime_omega_function]`_`f

• `F33f`_`[Multiplicative function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multiplicative_function]`_`f

• `F33f`_`[Arithmetic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arithmetic_function]`_`f

>>References

`:cite-note-erdos1939-1`!1.`! `F0af`_`[↑`#cite-ref-erdos1939-1-0]`_`f 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. online

>>Further reading

• Janko Bračič, `*Kolobar aritmetičnih funkcij`* (`*`F33f`_`[Ring`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ring_(algebra)]`_`f 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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