additive function
In number theory, an additive function
is an arithmetic function
with the property that and, for all with , .
An arithmetic function is said to be completely additive if and holds for all positive integers and , when they are not relatively prime. In this case, the function is a homomorphism
of monoids and, because of the fundamental theorem of arithmetic
, is completely determined by its restriction
to prime numbers
. Every completely additive function is additive.
Outside of number theory, the additive is usually used for all functions with the property for all arguments and . (For instance, see the other entry titled additive function (http://planetmath.org/AdditiveFunction2).) This entry discusses number theoretic additive functions.
Additive functions cannot have convolution inverses since an arithmetic function has a convolution inverse if and only if . A proof of this equivalence is supplied here (http://planetmath.org/ConvolutionInversesForArithmeticFunctions).
The most common of additive function in all of mathematics is the logarithm. Other additive functions that are useful in number theory are:
- •
-
•
, the number of (nondistinct) prime factors function (http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction)
By exponentiating an additive function, a multiplicative function is obtained. For example, the function is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function is completely multiplicative.
Title | additive function |
---|---|
Canonical name | AdditiveFunction1 |
Date of creation | 2013-03-22 16:07:03 |
Last modified on | 2013-03-22 16:07:03 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 12 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 11A25 |
Related topic | MultiplicativeFunction |
Defines | additive |
Defines | completely additive |
Defines | completely additive function |