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

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 ${\displaystyle f(n)}$ it is possible to create a related multiplicative function ${\displaystyle g(n),}$ which is a function with the property that whenever ${\displaystyle a}$ and ${\displaystyle b}$ are coprime then: $${\displaystyle g(ab)=g(a)\times g(b).}$$ One such example is ${\displaystyle g(n)=2^{f(n)}.}$ Likewise if ${\displaystyle f(n)}$ is completely additive, then ${\displaystyle g(n)=2^{f(n)}}$ is completely multiplicative. More generally, we could consider the function ${\displaystyle g(n)=c^{f(n)}}$, where ${\displaystyle c}$ is a nonzero real constant.

Given an additive function ${\displaystyle f}$, let its summatory function be defined by ${\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)}$. The average of ${\displaystyle f}$ 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 ${\displaystyle f}$ can be expanded as ${\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))}$ where $${\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}}$$