additive function
In number theory, an additive function
is an arithmetic function
f:ℕ→ℂ with the property that f(1)=0 and, for all a,b∈ℕ with gcd(a,b)=1, f(ab)=f(a)+f(b).
An arithmetic function f is said to be completely additive if f(1)=0 and f(ab)=f(a)+f(b) holds for all positive integers a and b, 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 f(a+b)=f(a)+f(b) for all arguments a and b. (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 f has a convolution inverse if and only if f(1)≠0. 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:
- •
-
•
Ω(n), 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 2ω(n) is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function 2Ω(n) 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 |