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In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions:[1]

f(ab) = f(a) + f(b).

Completely additive

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

Example of arithmetic functions which are completely additive are:

a0(4) = 2 + 2 = 4
a0(20) = a0(22 · 5) = 2 + 2+ 5 = 9
a0(27) = 3 + 3 + 3 = 9
a0(144) = a0(24 · 32) = a0(24) + a0(32) = 8 + 6 = 14
a0(2,000) = a0(24 · 53) = a0(24) + a0(53) = 8 + 15 = 23
a0(2,003) = 2003
a0(54,032,858,972,279) = 1240658
a0(54,032,858,972,302) = 1780417
a0(20,802,650,704,327,415) = 1240681
Ω(1) = 0, since 1 has no prime factors
Ω(4) = 2
Ω(16) = Ω(2·2·2·2) = 4
Ω(20) = Ω(2·2·5) = 3
Ω(27) = Ω(3·3·3) = 3
Ω(144) = Ω(24 · 32) = Ω(24) + Ω(32) = 4 + 2 = 6
Ω(2,000) = Ω(24 · 53) = Ω(24) + Ω(53) = 4 + 3 = 7
Ω(2,001) = 3
Ω(2,002) = 4
Ω(2,003) = 1
Ω(54,032,858,972,279) = 3
Ω(54,032,858,972,302) = 6
Ω(20,802,650,704,327,415) = 7

Example of arithmetic functions which are additive but not completely additive are:

ω(4) = 1
ω(16) = ω(24) = 1
ω(20) = ω(22 · 5) = 2
ω(27) = ω(33) = 1
ω(144) = ω(24 · 32) = ω(24) + ω(32) = 1 + 1 = 2
ω(2,000) = ω(24 · 53) = ω(24) + ω(53) = 1 + 1 = 2
ω(2,001) = 3
ω(2,002) = 4
ω(2,003) = 1
ω(54,032,858,972,279) = 3
ω(54,032,858,972,302) = 5
ω(20,802,650,704,327,415) = 5
a1(1) = 0
a1(4) = 2
a1(20) = 2 + 5 = 7
a1(27) = 3
a1(144) = a1(24 · 32) = a1(24) + a1(32) = 2 + 3 = 5
a1(2,000) = a1(24 · 53) = a1(24) + a1(53) = 2 + 5 = 7
a1(2,001) = 55
a1(2,002) = 33
a1(2,003) = 2003
a1(54,032,858,972,279) = 1238665
a1(54,032,858,972,302) = 1780410
a1(20,802,650,704,327,415) = 1238677

Multiplicative functions

From any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have:

g(ab) = g(a) × g(b).

One such example is g(n) = 2f(n).

Summatory functions

Given an additive function f, let its summatory function be defined by \( {\displaystyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)} \). The average of f is given exactly as

\( {\displaystyle {\mathcal {M}}_{f}(x)=\sum _{p^{\alpha }\leq x}f(p^{\alpha })\left(\left\lfloor {\frac {x}{p^{\alpha }}}\right\rfloor -\left\lfloor {\frac {x}{p^{\alpha +1}}}\right\rfloor \right).} \)

The summatory functions over f {\displaystyle f} f can be expanded as \( {\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))} \) where

\( {\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}} \)

The average of the function f 2 {\displaystyle f^{2}} f^{2} is also expressed by these functions as

\( {\displaystyle {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).} \)

There is always an absolute constant \( {\displaystyle C_{f}>0} \) such that for all natural numbers \( x \geq 1 \),

\( {\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).} \)

Let

\( {\displaystyle \nu (x;z):={\frac {1}{x}}\#\left\{n\leq x:{\frac {f(n)-A(x)}{B(x)}}\leq z\right\}.} \)

Suppose that f is an additive function with \( {\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1} \) such that as \( x\rightarrow \infty \) ,

\( {\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .} \)

Then \( {\displaystyle \nu (x;z)\sim G(z)} \) where G(z) is the Gaussian distribution function

\( {\displaystyle G(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}dt.} \)

Examples of this result related to the prime omega function and the numbers of prime divisors of shifted primes include the following for fixed \( {\displaystyle z\in \mathbb {R} } \) where the relations hold for \( {\displaystyle x\gg 1} \):

\( {\displaystyle \#\{n\leq x:\omega (n)-\log \log x\leq z(\log \log x)^{1/2}\}\sim xG(z),} \)
\( {\displaystyle \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).} \)

See also

Sigma additivity
Prime omega function
Multiplicative function
Arithmetic function

References

Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. online

Further reading

Janko Bračič, Kolobar aritmetičnih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25)
Iwaniec and Kowalski, Analytic number theory, AMS (2004).

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