Additive function: Difference between revisions

Revision as of 16:59, 26 January 2021

Revision as of 13:14, 1 July 2021

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{{shortShort description|Function that can be written as a sum over prime factors}}

{{About||the [[Abstract algebra|algebra]]ic meaning|Additive map}}

{{more footnotes|date=February 2013}}

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== Summatory functions ==

Given an additive function <math>f</math>, let its summatory function be defined by <math>\mathcal{M}_f(x) := \sum_{n \leq x} f(n)</math>. The average of <math>f</math> is given exactly as

:<math> \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). </math>

The summatory functions over <math>f</math> can be expanded as <math>\mathcal{M}_f(x) = x E(x) + O(\sqrt{x} \cdot D(x))</math> where

:<math> \begin{align}

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</math>

The average of the function <math>f^2</math> is also expressed by these functions as

:<math>\mathcal{M}_{f^2}(x) = x E^2(x) + O(x D^2(x)).</math>

There is always an absolute constant <math>C_f > 0</math> such that for all natural numbers <math>x \geq 1</math>,

:<math>\sum_{n \leq x} |f(n) - E(x)|^2 \leq C_f \cdot x D^2(x).</math>

Let

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

Suppose that <math>f</math> is an additive function with <math>-1 \leq f(p^{\alpha}) = f(p) \leq 1</math>

such that as <math>x \rightarrow \infty</math>,

:<math>B(x) = \sum_{p \leq x} f^2(p) / p \rightarrow \infty. </math>

Then <math>\nu(x; z) \sim G(z)</math> where <math>G(z)</math> is the [[normal distribution|Gaussian distribution function]]

:<math>G(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{-t^2/2} dt.</math>

Examples of this result related to the [[prime omega function]] and the numbers of prime divisors of shifted primes include the following for fixed <math>z \in \mathbb{R}</math> where the relations hold for <math>x \gg 1</math>:

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== Further reading ==

{{refbeginRefbegin}}

* Janko Bračič, ''Kolobar aritmetičnih funkcij'' (''[[Ring (algebra)|Ring]] of arithmetical functions''), (Obzornik mat, fiz. '''49''' (2002) 4, pp.&nbsp;97–108) <span style="color:darkblue;"> (MSC (2000) 11A25) </span>

* Iwaniec and Kowalski, ''Analytic number theory'', AMS (2004).

{{refendRefend}}

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[[Category:Arithmetic functions]]