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Concept
Arithmetic function
Summary
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".
An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.
There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes.
Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum.
An arithmetic function a is
completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n;
Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them.
Then an arithmetic function a is
additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;
multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n.
In this article, and mean that the sum or product is over all prime numbers:
and
Similarly, and mean that the sum or product is over all prime powers with strictly positive exponent (so k = 0 is not included):
The notations and mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if n = 12, then
The notations can be combined: and mean that the sum or product is over all prime divisors of n. For example, if n = 18, then
and similarly and mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then
The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ..
Official source
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