Arithmetic function

In number theory and computability theory, subfields of mathematics, a number-theoretic function is any function whose ___domain is the set of natural numbers.^[1]

A number-theoretic function whose range is included in the set of complex numbers is called an arithmetical function or arithmetic function.^[2] 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.

Formally, an arithmetic function is simply a sequence, with real or complex values. A sequence is, of course, a function on the set of natural numbers (i.e. positive integers). To emphasize that we are thinking of them as functions, we shall usually use notation like a(n), rather than a_n, for the value corresponding to the integer n. The term arithmetic function is used especially when a(n) is defined using number-theoretic properties in some way. A large part of number theory consists, in one way or another, of the study of these functions.

Examples

The articles on additive and multiplicative functions contain several examples of arithmetic functions. Here are some examples that are neither additive nor multiplicative:

• r₄(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 r₄(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).

• ω(n), the number of distinct primes dividing the number n. We have ω(1) = 0 and ω(20) = 2 (the distinct primes dividing 20 being 2 and 5).

• Λ(n), the von Mangoldt function which is defined to be ln(p) if n is an integer power of a prime p and 0 for all other n.

Footnotes

1. William J. LeVeque (1996). Fundamentals of Number Theory. Courier Dover Publications. ISBN 0486689069.
   Elliott Mendelson (1987). Introduction to Mathematical Logic. CRC Press. ISBN 0412808307.
2. Allan M. Kirch (1974). Elementary Number Theory: A Computer Approach. Intext Educational Publishers. ISBN 0700224564.
   R. Sivaramakrishnan and Sivaramakrishnan Sivaramakrishnan (1988). Classical Theory of Arithmetic Functions. Marcel Dekker. ISBN 0824780817.