Arithmetic functions

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, 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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