Additive function

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In number theory, an additive function is an arithmetic function ⁠${\displaystyle f}$⁠ on the natural numbers such that for all coprime ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠, $${\displaystyle f(ab)=f(a)+f(b).}$$ It follows that for any additive function, ${\displaystyle f(1)=0}$.

An additive function is said to be completely additive if ${\displaystyle f(ab)=f(a)+f(b)}$ holds for all ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠, even when they are not coprime. Totally additive is also used in this sense by analogy with totally multiplicative functions.

Examples

Multiplicative functions

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.

Summatory functions

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

The average of the function ${\displaystyle f^{2}}$ 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 ${\displaystyle x\geq 1}$, $${\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).}$$

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

Suppose that ${\displaystyle f}$ is an additive function with ${\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1}$ such that as ${\displaystyle x\rightarrow \infty }$, $${\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .}$$

Then ${\displaystyle \nu (x;z)\sim G(z)}$ where ${\displaystyle G(z)}$ 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

Further reading