arithmetic function

An arithmetic functionMathworldPlanetmath is a functionMathworldPlanetmath f:+ from the positive integers to the complex numbersPlanetmathPlanetmath.

Any algebraic functionMathworldPlanetmath over +, as well as transcendental functions such as sin(nπ) and enπi with n+ are arithmetic functions.

There are two noteworthy operations on the set of arithmetic functions:

If f and g are two arithmetic functions, the sum of f and g, denoted f+g, is given by


and the Dirichlet convolution of f and g, denoted by f*g, is given by


The set of arithmetic functions, equipped with these two binary operations, forms a commutative ring with unity. The 0 of the ring is the function f such that f(n)=0 for any positive integer n. The 1 of the ring is the function f with f(1)=1 and f(n)=0 for any n>1, and the units of the ring are those arithmetic function f such that f(1)0.

Note that giving a sequence {an} of complex numbers is equivalent to giving an arithmetic function by associating an with f(n).

Title arithmetic function
Canonical name ArithmeticFunction
Date of creation 2013-03-22 13:50:49
Last modified on 2013-03-22 13:50:49
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 10
Author mathcam (2727)
Entry type Definition
Classification msc 11A25
Related topic ConvolutionInversesForArithmeticFunctions
Related topic PropertyOfCompletelyMultiplicativeFunctions
Related topic DivisorSumOfAnArithmeticFunction
Defines Dirichlet convolution