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arithmetic function
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(Definition)
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An arithmetic function is a function $f:\Z^+\ra\mathbb{C}$ from the positive integers to the complex numbers.
Any algebraic function over $\Z^+$ , as well as transcendental functions such as $\sin(n\pi)$ and $e^{n\pi i}$ with $n\in \Z^+$ are arithmetic functions.
There are two noteworthy operations on the set of arithmetic functions:
If $f$ and $g$ are two arithmetic functions, the sum of $f$ and $g$ , denoted $f+g$ , is given by
and the Dirichlet convolution of $f$ and $g$ , denoted by $f*g$ , is given by
The set of arithmetic functions, equipped with these two binary operations, forms a commutative ring with unity. 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 $n>1$ , and the units of the ring
are those arithmetic function $f$ such that $f(1)\neq 0$ .
Note that giving a sequence $\{a_n\}$ of complex numbers is equivalent to giving an arithmetic function by associating $a_n$ with $f(n)$ .
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Cross-references: equivalent, sequence, units, ring, unity, commutative ring, binary operations, sum, operations, transcendental functions, algebraic function, complex numbers, integers, positive, function
There are 17 references to this entry.
This is version 7 of arithmetic function, born on 2003-08-14, modified 2008-10-24.
Object id is 4584, canonical name is ArithmeticFunction.
Accessed 9427 times total.
Classification:
| AMS MSC: | 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas) |
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Pending Errata and Addenda
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