Online Encyclopedia
Arithmetic function
In number theory, an arithmetic function (or numbertheoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. In other words: an arithmetic function is nothing but a sequence of complex numbers.
The most important arithmetic functions are the additive and the multiplicative ones.
An important operation on arithmetic functions is the Dirichlet convolution.
Examples
The articles on additive and multiplicative functions contain several examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:
 c_{4}(n)  the number of ways that n can be expressed as the sum of four squares of nonnegative integers, where we distinguish between different orders of the summands. For example:

 1 = 1^{2}+0^{2}+0^{2}+0^{2} = 0^{2}+1^{2}+0^{2}+0^{2} = 0^{2}+0^{2}+1^{2}+0^{2} = 0^{2}+0^{2}+0^{2}+1^{2},
 hence c_{4}(1)=4.
 P(n), the Partition function  the number of representations of n as a sum of positive integers, where we don't distinguish between different orders of the summands. For instance: P(2 · 5) = P(10) = 42 and P(2)P(5) = 2 · 7 = 14 ≠ 42.
 π (n), the Prime counting function  the number of primes less than or equal to a given number n. We have π(1) = 0 and π(10) = 4 (the primes below 10 being 2, 3, 5, and 7).
Last updated: 10242004 05:10:45