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Function that can be written as a sum over prime factors From Wikipedia, the free encyclopedia
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b:[1]
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (February 2013) |
An additive function f(n) is said to be completely additive if holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0.
Every completely additive function is additive, but not vice versa.
Examples of arithmetic functions which are completely additive are:
Examples of arithmetic functions which are additive but not completely additive are:
From any additive function it is possible to create a related multiplicative function which is a function with the property that whenever and are coprime then: One such example is Likewise if is completely additive, then is completely multiplicative. More generally, we could consider the function , where is a nonzero real constant.
Given an additive function , let its summatory function be defined by . The average of is given exactly as
The summatory functions over can be expanded as where
The average of the function is also expressed by these functions as
There is always an absolute constant such that for all natural numbers ,
Let
Suppose that is an additive function with such that as ,
Then where is the Gaussian distribution function
Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed where the relations hold for :
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