Additive function: Difference between revisions

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:''g''(''ab'') = ''g''(''a'') × ''g''(''b'').

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

== 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}

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{align}

</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>

== See also ==

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{{refbegin}}

* 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).

{{refend}}