Arithmetic function

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function f(n) defined on the set of natural numbers (i.e., positive integers) that "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, e.g. the prime-counting functions. This article provides links to functions of both classes.

${\displaystyle \sum _{p}f(p)\;}$   and   ${\displaystyle \prod _{p}f(p)\;}$   mean that the sum or product is over all prime numbers:

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