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] f ( a b ) = f ( a ) + f ( b ) . {\displaystyle f(ab)=f(a)+f(b).}
An additive function f(n) is said to be completely additive if f ( a b ) = f ( a ) + f ( b ) {\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 f ( n ) {\displaystyle f(n)} it is possible to create a related multiplicative function g ( n ) , {\displaystyle g(n),} which is a function with the property that whenever a {\displaystyle a} and b {\displaystyle b} are coprime then: g ( a b ) = g ( a ) × g ( b ) . {\displaystyle g(ab)=g(a)\times g(b).} One such example is g ( n ) = 2 f ( n ) . {\displaystyle g(n)=2^{f(n)}.} Likewise if f ( n ) {\displaystyle f(n)} is completely additive, then g ( n ) = 2 f ( n ) {\displaystyle g(n)=2^{f(n)}} is completely multiplicative. More generally, we could consider the function g ( n ) = c f ( n ) {\displaystyle g(n)=c^{f(n)}} , where c {\displaystyle c} is a nonzero real constant.
Given an additive function f {\displaystyle f} , let its summatory function be defined by M f ( x ) := ∑ n ≤ x f ( n ) {\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)} . The average of f {\displaystyle f} is given exactly as M f ( x ) = ∑ p α ≤ x f ( p α ) ( ⌊ x p α ⌋ − ⌊ x p α + 1 ⌋ ) . {\displaystyle {\mathcal {M}}_{f}(x)=\sum _{p^{\alpha }\leq x}f(p^{\alpha })\left(\left\lfloor {\frac {x}{p^{\alpha }}}\right\rfloor -\left\lfloor {\frac {x}{p^{\alpha +1}}}\right\rfloor \right).}
The summatory functions over f {\displaystyle f} can be expanded as M f ( x ) = x E ( x ) + O ( x ⋅ D ( x ) ) {\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))} where E ( x ) = ∑ p α ≤ x f ( p α ) p − α ( 1 − p − 1 ) D 2 ( x ) = ∑ p α ≤ x | f ( p α ) | 2 p − α . {\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}}
The average of the function f 2 {\displaystyle f^{2}} is also expressed by these functions as M f 2 ( x ) = x E 2 ( x ) + O ( x D 2 ( x ) ) . {\displaystyle {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).}
There is always an absolute constant C f > 0 {\displaystyle C_{f}>0} such that for all natural numbers x ≥ 1 {\displaystyle x\geq 1} , ∑ n ≤ x | f ( n ) − E ( x ) | 2 ≤ C f ⋅ x D 2 ( x ) . {\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).}
Let ν ( x ; z ) := 1 x # { n ≤ x : f ( n ) − A ( x ) B ( x ) ≤ z } . {\displaystyle \nu (x;z):={\frac {1}{x}}\#\!\left\{n\leq x:{\frac {f(n)-A(x)}{B(x)}}\leq z\right\}\!.}
Suppose that f {\displaystyle f} is an additive function with − 1 ≤ f ( p α ) = f ( p ) ≤ 1 {\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1} such that as x → ∞ {\displaystyle x\rightarrow \infty } , B ( x ) = ∑ p ≤ x f 2 ( p ) / p → ∞ . {\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .}
Then ν ( x ; z ) ∼ G ( z ) {\displaystyle \nu (x;z)\sim G(z)} where G ( z ) {\displaystyle G(z)} is the Gaussian distribution function G ( z ) = 1 2 π ∫ − ∞ z e − t 2 / 2 d t . {\displaystyle G(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}dt.}
Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed z ∈ R {\displaystyle z\in \mathbb {R} } where the relations hold for x ≫ 1 {\displaystyle x\gg 1} : # { n ≤ x : ω ( n ) − log log x ≤ z ( log log x ) 1 / 2 } ∼ x G ( z ) , {\displaystyle \#\{n\leq x:\omega (n)-\log \log x\leq z(\log \log x)^{1/2}\}\sim xG(z),} # { p ≤ x : ω ( p + 1 ) − log log x ≤ z ( log log x ) 1 / 2 } ∼ π ( x ) G ( z ) . {\displaystyle \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).}