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.
Examples
The articles on additive and multiplicative functions contain several important examples. Examples of a non-multiplicative functions are:
- 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 ≠ 1.
- 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 ≠ 1, π(2 · 5) = π(10) = 4 and π(2) π(5) = 1 · 3 = 3 ≠ 4.
- a0(n) - the sum of primes dividing n, sometimes called sopfr(n). We have a0(1) = 0 ≠ 1, a0(2 · 5) = a0(10) = 7 and a0(2) a0(5) = 2 · 5 = 10 ≠ 7(SIDN A001414).
- a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n). We have a1(1) = 0 ≠ 1, a1(2 · 5) = a1(10) = 7 and a1(2) a1(5) = 2 · 5 = 10 ≠ 7(SIDN A008472).