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| 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 $\gcd(a,b)=1$, $f(ab)=f(a)+f(b)$. |
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 $\gcd(a,b)=1$, $f(ab)=f(a)+f(b)$. |
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| An arithmetic function $f$ is said to be {\sl completely additive\/} if $f(1)=0$ and $f(ab)=f(a)+f(b)$ holds for {\sl all\/} positive integers $a$ and $b$, \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 prime numbers. Every completely additive function is additive. |
An arithmetic function $f$ is said to be {\sl completely additive\/} if $f(1)=0$ and $f(ab)=f(a)+f(b)$ holds for {\sl all\/} positive integers $a$ and $b$, \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 prime numbers. Every completely additive function is additive. |
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| Outside of number theory, the \PMlinkescapetext{term} 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 \PMlinkname{additive function}{AdditiveFunction2}.) This entry discusses number theoretic additive functions. |
Outside of number theory, the \PMlinkescapetext{term} 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 \PMlinkname{additive function}{AdditiveFunction2}.) This entry discusses number theoretic additive functions. |
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| Additive functions cannot have convolution inverses since an arithmetic function $f$ has a convolution inverse if and only if $f(1) \neq 0$. A proof of this equivalence is supplied \PMlinkname{here}{ConvolutionInversesForArithmeticFunctions}. |
Additive functions cannot have convolution inverses since an arithmetic function $f$ has a convolution inverse if and only if $f(1) \neq 0$. A proof of this equivalence is supplied \PMlinkname{here}{ConvolutionInversesForArithmeticFunctions}. |
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| The most common \PMlinkescapetext{type} of additive function in all of mathematics is the logarithm. Other additive functions that are useful in number theory are: |
The most common \PMlinkescapetext{type} of additive function in all of mathematics is the logarithm. Other additive functions that are useful in number theory are: |
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| \begin{itemize} |
\begin{itemize} |
| \item $\omega(n)$, the number of distinct prime factors function |
\item $\omega(n)$, the number of distinct prime factors function |
| \item $\Omega(n)$, the \PMlinkname{number of (nondistinct) prime factors function}{NumberOfNondistinctPrimeFactorsFunction} |
\item $\Omega(n)$, the \PMlinkname{number of (nondistinct) prime factors function}{NumberOfNondistinctPrimeFactorsFunction} |
| \end{itemize} |
\end{itemize} |
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By exponentiating an additive function, a multiplicative function is obtained. For example, the function $\displaystyle 2^{\omega(n)}$ is multiplicative. Similarly, by exponentiating a completely additive function, a completely multiplicative function is obtained. For example, the function $\displaystyle 2^{\Omega(n)}$ is completely multiplicative.
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By exponentiating an additive function, a multiplicative function is obtained. For example, the function $\displaystyle 2^{\omega(n)}$ is multiplicative. multiplicative.
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