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.

Arithmetic functions may be studied with Bell series.

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).