Formally, in number theory, 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 an, 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.[1]
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
- ^ G. J. O. Jameson (2003). The Prime Number Theorem. London Mathematical Society. ISBN 0-521-89110-8.