Additive function
In mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 the term additive function has two different definitions, depending on the specific field of application.

In algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 an additive function (or additive map) 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. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation
Cauchy's functional equation
Cauchy's functional equation is the functional equation f=f+f. \ Solutions to this are called additive functions.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely f = cx \ for any arbitrary rational number c.Over the real...


In number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, an additive function is an arithmetic function
Arithmetic function
In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...

 f(n) of the positive integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 n such that whenever a and b are coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

, 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
Additive polynomial
In mathematics, the additive polynomials are an important topic in classical algebraic number theory.-Definition:Let k be a field of characteristic p, with p a prime number. A polynomial P with coefficients in k is called an additive polynomial, or a Frobenius polynomial, ifP=P+P\,as polynomials...

. Note also that any homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 f between Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

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 co-prime. 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.


Example of arithmetic functions which are completely additive are:
  • The restriction of the logarithmic function
    The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

     to N.

  • The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n.

  • a0(n) - the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n . 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(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
a0(2,003) = 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
    Prime number
    A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

     factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" . For example;

Ω(1) = 0, since 1 has no prime factors
Ω(20) = Ω(2·2·5) = 3
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

Example of arithmetic functions which are additive but not completely additive are:
  • ω(n), defined as the total number of different prime factors of n . For example:

ω(4) = 1
ω(20) = ω(22·5) = 2
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

  • a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) . 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(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
a1(2,001) = 55
a1(2,002) = 33
a1(2,003) = 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 easy to create a related multiplicative function
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...

 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) = 2f(n).

Further reading

  • Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)
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