Average order of an arithmetic function: Difference between revisions

Revision as of 10:57, 10 August 2014

Revision as of 11:41, 10 August 2014

Line 281:

'''Proof.'''

Let ''m'' be a monic polynomial, and suppose that <math>m=\prod_{i \mathop =1}^{l}P_{i}^{e_i}</math> where <math>P_i</math> are the prime factors of ''m''.

First, notice that,

We have,

<math>\sum_{f|m}\Lambda_{A}(f)=log|m|</math>.

<math>

\begin{align}

<math>\sum_{f|m}\Lambda_{A}(f)=log|m|</math>.

&= \sum_{(i_1,...,i_l)|0\le i_{j} \le e_j}\Lambda_{A}(\prod_{j \mathop =1}^{l}P_{j}^{i_j})=\sum_{j \mathop =1}^{l}\sum_{i \mathop =1}^{e_i}\Lambda_{A}(P_{j}^i)=\sum_{j \mathop =1}^{l}\sum_{i \mathop =1}^{e_i}\log|P_j|\\

&= \sum_{j \mathop =1}^{l}e_j\log|P_j| =\sum_{j \mathop =1}^{l}\log|P_j|^{e_j}=\log|(\prod_{i \mathop =1}^{l}P_{i}^{e_i})|\\

&= \log(m)

\end{align}

</math>

Hence,