In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.
Definition
A completely multiplicative function is an arithmetic function, such that f = 1 and f = ff holds for allpositive integers a and b. Without the requirement that f = 1, one could still have f = 0, but then f = 0 for all positive integers a, so this is not a very strong restriction. The definition above can be rephrased using the language of algebra: A completely multiplicative function is a homomorphism from the monoid to some other monoid.
A completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = paqb..., then f = fafb... While the Dirichlet convolution of two multiplicative functions is multiplicative, the Dirichlet convolution of two completely multiplicative functions need not be completely multiplicative. There are a variety of statements about a function which are equivalent to it being completely multiplicative. For example, if a function f is multiplicative then it is completely multiplicative if and only if its Dirichlet inverse is where is the Möbius function. Completely multiplicative functions also satisfy a distributive law. If f is completely multiplicative then where * represents the Dirichlet product and represents pointwise multiplication. One consequence of this is that for any completely multiplicative function f one has which can be deduced from the above by putting both, where is the constant function. Here is the divisor function.
The L-function of completely multiplicative Dirichlet series satisfies which means that the sum all over the natural numbers is equal to the product all over the prime numbers.
