Arithmetic function

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'. 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. Many of the functions mentioned in this article have expansions as series involving these sums; see the article Ramanujan's sum for examples. An arithmetic function a is Two whole numbers m and n are called coprime if their greatest common divisor is 1; i.e., if there is no prime number that divides both of them. Then an arithmetic function a is ∑ p f ( p ) {displaystyle sum _{p}f(p)}   and   ∏ p f ( p ) {displaystyle prod _{p}f(p)}   mean that the sum or product is over all prime numbers: Similarly,   ∑ p k f ( p k ) {displaystyle sum _{p^{k}}f(p^{k})}   and   ∏ p k f ( p k ) {displaystyle prod _{p^{k}}f(p^{k})}   mean that the sum or product is over all prime powers with strictly positive exponent (so 1 is not included):