arithmetic function (original) (raw)

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).