- Additive function
Different definitions exist depending on the specific field of application. Traditionally, an

**additive function**is a function that preserves the addition operation::"f"("x" + "y") = "f"("x") + "f"("y")for any two elements "x" and "y" in the domain. An example of an additive function would include the total-deriviate operator d; that is to say d(x + y) = d(x) + d(y).In

number theory , an**additive function**is anarithmetic function "f"("n") of the positiveinteger "n" such that whenever "a" and "b" arecoprime , the function of the product is the sum of the functions::"f"("ab") = "f"("a") + "f"("b").The remainder of this article discusses number theoretic additive functions, using the second definition.For a specific case of the first definition see

additive polynomial . Note also that anyhomomorphism "f" betweenAbelian group s is "additive" by the first definition.**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 withtotally multiplicative functions.Every completely additive function is additive, but not vice versa.

**Examples**Arithmetic functions which are completely additive are:

* The restriction of the logarithmic function to**N*** "a"

_{0}("n") - the sum of primes dividing "n", sometimes called sopfr("n"). We have "a"_{0}(20) = "a"_{0}(2^{2}· 5) = 2 + 2+ 5 = 9. Some values: ( [*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001414 OEIS A001414*] ).::"a"

_{0}(4) = 4::"a"_{0}(27) = 9::"a"_{0}(144) = "a"_{0}(2^{4}· 3^{2}) = "a"_{0}(2^{4}) + "a"_{0}(3^{2}) = 8 + 6 = 14::"a"_{0}(2,000) = "a"_{0}(2^{4}· 5^{3}) = "a"_{0}(2^{4}) + "a"_{0}(5^{3}) = 8 + 15 = 23::"a"_{0}(2,003) = 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 prime factors of "n", counting multiple factors multiple times. It is often called "

Big Omega function ".This implies Ω(1) = 0 since 1 has no prime factors. Some more values: ( [*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001222 OEIS A001222*] )::Ω(4) = 2::Ω(27) = 3::Ω(144) = Ω(2

^{4}· 3^{2}) = Ω(2^{4}) + Ω(3^{2}) = 4 + 2 = 6::Ω(2,000) = Ω(2^{4}· 5^{3}) = Ω(2^{4}) + Ω(5^{3}) = 4 + 3 = 7::Ω(2,001) = 3::Ω(2,002) = 4::Ω(2,003) = 1::Ω(54,032,858,972,279) = 3::Ω(54,032,858,972,302) = 6 ::Ω(20,802,650,704,327,415) = 7:: ...* The function "a"

_{1}("n") - the sum of the distinct primes dividing "n", sometimes called sopf("n"), is additive but not completely additive. We have "a"_{1}(1) = 0, "a"_{1}(20) = 2 + 5 = 7. Some more values: ( [*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A008472 OEIS A008472*] )::"a"

_{1}(4) = 2::"a"_{1}(27) = 3::"a"_{1}(144) = "a"_{1}(2^{4}· 3^{2}) = "a"_{1}(2^{4}) + "a"_{1}(3^{2}) = 2 + 3 = 5::"a"_{1}(2,000) = "a"_{1}(2^{4}· 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) = 1238665 ::"a"_{1}(54,032,858,972,302) = 1780410 ::"a"_{1}(20,802,650,704,327,415) = 1238677:: ...* Another example of an arithmetic function which is additive but not completely additive is ω("n"), defined as the total number of "different" prime factors of "n". Some values (compare with Ω("n")) ( [

*http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001221 OEIS A001221*] ):::ω(4) = 1::ω(27) = 1::ω(144) = ω(2

^{4}· 3^{2}) = ω(2^{4}) + ω(3^{2}) = 1 + 1 = 2::ω(2,000) = ω(2^{4}· 5^{3}) = ω(2^{4}) + ω(5^{3}) = 1 + 1 = 2::ω(2,001) = 3::ω(2,002) = 4::ω(2,003) = 1::ω(54,032,858,972,279) = 3::ω(54,032,858,972,302) = 5 ::ω(20,802,650,704,327,415) = 5:: ...**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") × "g"("b").One such example is "g"("n") = 2^{("f"("n")-"f"(1))}.**References**# Janko Bračič, "Kolobar aritmetičnih funkcij" ("Ring of arithmetical functions"), (Obzornik mat, fiz.

**49**(2002) 4, pp 97 - 108) (MSC (2000) 11A25)**See also***

Sigma additivity

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**additive function**— noun A unary function that preserves the addition operation; a unary function f such that for any x and y: f(x + y) = f(x) + f(y) … Wiktionary**countably additive function**— Math. a set function that upon operating on the union of a countable number of disjoint sets gives the same result as the sum of the functional values of each set. Cf. finitely additive function. * * * … Universalium**finitely additive function**— Math. a set function that upon operating on the union of a finite number of disjoint sets gives the same result as the sum of the functional values of each set. Cf. countably additive function. * * * … Universalium**countably additive function**— Math. a set function that upon operating on the union of a countable number of disjoint sets gives the same result as the sum of the functional values of each set. Cf. finitely additive function … Useful english dictionary**finitely additive function**— Math. a set function that upon operating on the union of a finite number of disjoint sets gives the same result as the sum of the functional values of each set. Cf. countably additive function … Useful english dictionary**Additive**— may refer to:* Additive function, a function which preserves addition * Additive inverse, an arithmetic concept * Additive category, a preadditive category with finite biproducts * Additive rhythm, a larger period of time constructed from smaller … Wikipedia**Additive synthesis**— is a technique of audio synthesis which creates musical timbre.The timbre of an instrument is composed of multiple harmonics or partials , in different quantities, that change over time. Additive synthesis emulates such timbres by combining… … Wikipedia**Additive inverse**— In mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero. The additive inverse of a is denoted −a. For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive… … Wikipedia**additive**— additively, adv. /ad i tiv/, n. 1. something that is added, as one substance to another, to alter or improve the general quality or to counteract undesirable properties: an additive that thins paint. 2. Nutrition. a. Also called food additive. a… … Universalium**Additive identity**— In mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x . One of the most familiar additive identities is the number 0 from elementary… … Wikipedia