Average order of an arithmetic function: Difference between revisions

Revision as of 09:43, 10 August 2014

Revision as of 10:21, 10 August 2014

Line 229:

<math>\text{LHS}=\sum_{n}(\sum_{\text{deg}(m)=n} \sigma_k(m))u^{n}</math>, and by [[Cauchy product]] we get,

<math>

<math>\text{RHS}=\sum_{n}(q^{n}(\frac{1-q^{k(n+1)}}{1-q^{k}}))u^{n}</math>.

\begin{align}

\text{RHS}

&=\sum_{n}q^{n(1-s)}\sum_{n}(\sum_{\text{deg}(m)=n}h(m))u^{n}\\

&=\sum_{n}q^{n}u^{n}\sum_{n}q^{n}q^{nk}u^{n}\\

&=\sum_{n}(\sum_{j \mathop =0}^{n}q^{n-j}q^{jk+j})\\

<math>\text{RHS}&=\sum_{n}(q^{n}(\frac{1-q^{k(n+1)}}{1-q^{k}}))u^{n}</math>.

\end{align}

</math>.

Finally we get that,