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arithmetic function (Definition)

An arithmetic function is a function $ f:\mathbb{Z}^+\rightarrow \mathbb{C}$ from the positive integers to the complex numbers.

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

$\displaystyle (f+g)(n)=f(n)+g(n),$    

and the Dirichlet convolution of $ f$ and $ g$, denoted by $ f*g$, is given by
$\displaystyle (f*g)(n)=\sum_{d\vert n}f(d)g\left(\frac{n}{d}\right).$    

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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See Also: convolution inverses for arithmetic functions, pointwise multiplication of a completely multiplicative function distibutes over convolution, divisor sum of an arithmetic function

Also defines:  Dirichlet convolution

Attachments:
arithmetic functions form a ring (Theorem) by rm50
divisor sum of an arithmetic function (Definition) by azdbacks4234
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Cross-references: equivalent, sequence, units, ring, unity, commutative ring, binary operations, sum, operations, complex numbers, integers, positive, function
There are 14 references to this entry.

This is version 6 of arithmetic function, born on 2003-08-14, modified 2006-06-12.
Object id is 4584, canonical name is ArithmeticFunction.
Accessed 6415 times total.

Classification:
AMS MSC: 11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)
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