Additive function

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]

{\displaystyle f(ab)=f(a)+f(b).}

An additive function f(n) is said to be completely additive if

{\displaystyle f(ab)=f(a)+f(b)}

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

is completely additive, then

is completely multiplicative.

More generally, we could consider the function

is a nonzero real constant.

Given an additive function

, let its summatory function be defined by

The summatory functions over

The average of the function

is also expressed by these functions as

There is always an absolute constant

such that for all natural numbers

ν ( x ; z ) :=

is an additive function with

ν ( x ; z ) ∼

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

# { n ≤ x : ω ( n ) − log ⁡ log ⁡ x ≤ z ( log ⁡ log ⁡ x

# { p ≤ x : ω ( p + 1 ) − log ⁡ log ⁡ x ≤ z ( log ⁡ log ⁡ x