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I try to separate the following double sum

$$S=\sum_{i,j \in[-n,n], (i,j)\ne (0,0)} f_{ij}$$

where

$$f_{ij}=\frac{(-1)^{j+k}}{\sqrt{j^2+k^2}}$$

$$S=\sum _{k=1}^n \sum _{j=1}^n (f(-k,-j)+f(-k,j)+f(k,-j)+f(k,j))$$

but I do not know why it does not work for several value of integers.

Could you help to get a formula for n dimensional sum

Thanks

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  • $\begingroup$ I try sum lattice sums and I need a general n dimension formula ,it is a way of begining ,Thanks @Jean Marie $\endgroup$ May 16, 2021 at 8:10
  • $\begingroup$ Your title is misleading: as I understand, you look for a closed-form formula... $\endgroup$
    – Jean Marie
    May 16, 2021 at 15:32
  • $\begingroup$ Yes pleases , in dimension two or three is will be ok $\endgroup$ May 16, 2021 at 16:29
  • $\begingroup$ What do you mean by "change of indices", by "separate" ? Explain with detailed sentences what you want to do. $\endgroup$
    – Jean Marie
    May 16, 2021 at 20:43
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    $\begingroup$ Your last expression is not the right one because you exclude for example indices (k=1,j=0) which should be included. $\endgroup$
    – Jean Marie
    May 17, 2021 at 8:09

1 Answer 1

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Here is a numerical simulation giving the values of $S$ from $n=1$ to $n=100$.

A convergent behavior takes place with oscillations around a limit $\approx 1.164$.

enter image description here

but I am unable to find its exact value.

Nevertheless, this kind of sum has been studied: it is $b_2(1)$ with $b_2(s)$ given by formula (3), Problem n° 3 in this interesting [document] (https://www.davidhbailey.com//dhbpapers/tenproblems.pdf)

See as well here.

Both documents mention this formula:

$$b_2(s)=-4\beta(s)\zeta(s)$$

which unfortunately is valid only for $s>1$.

Somewhat related: Infinite sum involving number of solutions to $k=i^2+j^2$

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  • $\begingroup$ thanks @Jean Marie for the try maybe I express myself really bad what I want , I can calculation de numerical result but this is not the question, the double sum goes fron - Infinity t o infnity I need it is to get a representation of this sum for 0 to infnity for other reasons $\endgroup$ May 17, 2021 at 12:34
  • $\begingroup$ Could you say what you call "to get a representation" ? Is it a "closed-form formula" (see what this expression mean) ? And what do you mean about "n" dimensional sum in your question ? $\endgroup$
    – Jean Marie
    May 17, 2021 at 12:36
  • $\begingroup$ in Math languaje $$\sum _{k=-n}^n \sum _{j=-n}^n \to \sum _{k=1}^n \sum _{j=1}^n$$ $\endgroup$ May 17, 2021 at 18:22
  • $\begingroup$ You had asked a question previously a question on Madelung constant : you should link your questions in order to help people to help you. $\endgroup$
    – Jean Marie
    May 20, 2021 at 20:53

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