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# Arithmetic function

In number theory, an arithmetic function (or number-theoretic 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:

• c4(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 = 12+02+02+02 = 02+12+02+02 = 02+02+12+02 = 02+02+02+12,
hence c4(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: 10-24-2004 05:10:45