| Version 3 |
Version 2 |
| An \emph{arithmetic function} is a function $f:\Z^+\ra\C$ from the positive integers to the complex numbers. |
An \emph{arithmetic function} is a function $f:\Z^+\ra\C$ from the positive integers to the complex numbers. |
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| There are two noteworthy operations on the set of arithmetic functions: |
There are two noteworthy operations on the set of arithmetic functions: |
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| If $f$ and $g$ are two arithmetic functions, the \emph{sum} of $f$ and $g$, denoted $f+g$, is given by |
If $f$ and $g$ are two arithmetic functions, the \emph{sum} of $f$ and $g$, denoted $f+g$, is given by |
| \begin{align*} |
\begin{align*} |
| (f+g)(n)=f(n)+g(n), |
(f+g)(n)=f(n)+g(n), |
| \end{align*} |
\end{align*} |
| and the \emph{Dirichlet convolution} of $f$ and $g$, denoted by $f*g$, is given by |
and the \emph{Dirichlet convolution} of $f$ and $g$, denoted by $f*g$, is given by |
| \begin{align*} |
\begin{align*} |
| (f*g)(n)=\sum_{d|n}f(d)g\left(\frac{n}{d}\right). |
(f*g)(n)=\sum_{d|n}f(d)g\left(\frac{n}{d}\right). |
| \end{align*} |
\end{align*} |
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The set or arithmetic functions, equipped with these two binary operations, forms a commutative ring. The 0 of the ring is the function $f$ such that $f(n)=0$ for any positive integer $n$. The 1 of the ring is the function $f$ with $f(1)=1$ and $f(n)=0$ for any $n>1$. |
| The set of arithmetic functions, equipped with these two binary operations, forms a commutative ring. The 0 of the ring is the function $f$ such that $f(n)=0$ for any positive integer $n$. The 1 of the ring is the function $f$ with $f(1)=1$ and $f(n)=0$ for any $n>1$. |
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