I was reading a proof of “Generalised Convolution” in Tom Apostol’s Analytical Number Theory. I seem to have lost my mind in trying to figure out how the switch in the summation indices is performed (page 39 of the book)
The equation goes like this:
$$\sum_{n\leq x}\alpha(n) \sum_{m\leq \frac{x}{n}}\beta(m)F( \frac{x}{mn}) $$ And switching it to this is (confusing me part 1): $$\sum_{mn\leq x}\alpha(n)\beta(m)F(\frac{x}{mn}) $$ The above equation is again switched to (confusing me part 2): $$\sum_{k\leq x}\left(\sum_{n|k} \alpha(n) \beta(k/n)\right)F(\frac{x}{k}) $$ Please help.
It might be helpful to write the index region with some additional information. We obtain \begin{align*} \color{blue}{\sum_{1\leq n\leq x}}&\color{blue}{\alpha(n)\sum_{1\leq m\leq \frac{x}{n}}\beta(m)F\left(\frac{x}{mn}\right)}\\ &=\sum_{{1\leq n\leq nm\leq x}\atop{1\leq m}}\alpha(n)\beta(m)F\left(\frac{x}{mn}\right)\tag{1}\\ &=\sum_{{1\leq n\leq k\leq x}\atop{n\mid k}}\alpha(n)\beta(k/n)F\left(\frac{x}{k}\right)\tag{2}\\ &\,\,\color{blue}{=\sum_{1\leq k\leq x}\left(\sum_{{n\mid k}\atop{1\leq n}}\alpha(n)\beta(k/n)\right)F\left(\frac{x}{k}\right)} \end{align*} according to the claim. We observe in the last line we have just exchanged inner and outer sum of the first line.
Comment:
In (1) we write the index region as inequality chain, multiplying the index region of the inner sum with $n$.
In (2) we substitute $k=m n$.