In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
, an additive function is an
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
''f''(''n'') of the positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
variable ''n'' such that whenever ''a'' and ''b'' are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...
, the function applied to the product ''ab'' is the sum of the values of the function applied to ''a'' and ''b'':
[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]
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Completely additive
An additive function ''f''(''n'') is said to be completely additive if 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 In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
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 logarithmic function to
* The multiplicity of a prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way