I started solving a problem using the transformation of Reimann zeta function into this form:

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And I searched for the methods of doing this transformation and ended up with this one :

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It is a little bit unclear to me the way it exchanges the summation signs , the first summation changes variables from $n \geq 1 $ to $k \geq 1$ :

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Any explanation to this , or any help on how to transform Reimann Zeta function into that form(using other methods) would be appreciated, thank you.

  • 1
    $\begingroup$ There is nothing strange in this, it's just an interchange of the two sums. If you notice, the first one says "$n$ is greater or equal to $1$ and $k$ must be equal or greater to $n$". Then we may change it into "$k$ is greater or equal to $1$ provided that $n$ is equal or less than $k$". Simple reasoning. $\endgroup$ Dec 11 '18 at 9:55
  • $\begingroup$ @VonNeumann so are this to expressions $\sum_{n=1}^{\infty} \sum_{k=n}^{\infty}$ and $\sum_{k=1}^{\infty} \sum_{n=k}^{\infty}$ equal , I mean am I allowed to do these kinda exchanges ? $\endgroup$ Dec 11 '18 at 9:59
  • $\begingroup$ You need absolute convergence of the double-series in order to do this. (This is fulfilled here.) $\endgroup$
    – p4sch
    Dec 11 '18 at 10:24

Something that works very nice to see the change of indexes is writing a sum like this

$$\sum_{\color{red}{n}\ge 1}\sum_{\color{green}{k}\ge n}=\sum_{1\le\color{red}{ n}<\infty}\sum_{n\le \color{green}{k}<\infty}=\sum_{1\le \color{red}{n}\le \color{green}{k}<\infty}$$

where I use colors to denote the variables clearly in each sum. The notation at the LHS seems a bit sloppy, it is writing $k\ge n$ where $k$ is the variable, not $n$. Now choosing $k$ as a variable from $1$ to infinity we get

$$\sum_{1\le \color{red}{n}\le\color{green}{k}<\infty}=\sum_{1\le\color{green}{ k}\le\infty}\sum_{1\le \color{red}{n}\le k}=\sum_{k=1}^\infty\sum_{n=1}^k$$

where the last expression is equivalent to the sloppy form $\sum_{k\ge 1}\sum_{n\le k}$. This reordering is justified when the double sum is summable, this happen, by example, when it is a double sum of non-negative terms or when it is absolutely convergent.


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