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| Title of object: |
additive function |
| Canonical Name: |
AdditiveFunction |
| Type: |
Definition |
| Created on: |
2006-07-27 19:41:39 |
| Modified on: |
2006-07-27 19:41:39 |
| Classification: |
msc:11A25 |
| Defines: |
additive, completely additive, completely additive function |
Preamble:
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Content:
In number theory, an {\sl additive function\/} is an arithmetic function $f \colon \mathbb{N} \to \mathbb{C}$ with the property that $f(1)=0$ and, for all $a,b \in \mathbb{N}$ 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 $a,b \in \mathbb{N}$, \PMlinkescapetext{even} 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 the prime numbers. Every completely additive function is additive.
The most common \PMlinkescapetext{type} of additive function in all of mathematics is the logarithm. Other common additive functions that are useful in number theory are:
\begin{itemize}
\item $\omega(n)$, the number of distinct prime factors function
\item $\Omega(n)$, the \PMlinkname{number of (nondistinct) prime factors function}{NumberOfNondistinctPrimeFactorsFunction}
\end{itemize}
By exponentiating an additive function, a multiplicative function is obtained. For example, the function $\displaystyle 2^{\omega(n)}$ is multiplicative.
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