Additive function

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 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").

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 any homomorphism "f" between Abelian groups 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 with totally 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(22 · 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(24 · 32) = "a"0(24) + "a"0(32) = 8 + 6 = 14::"a"0(2,000) = "a"0(24 · 53) = "a"0(24) + "a"0(53) = 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(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6::Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 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(24 · 32) = "a"1(24) + "a"1(32) = 2 + 3 = 5::"a"1(2,000) = "a"1(24 · 53) = "a"1(24) + "a"1(53) = 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(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2::ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 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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