Expressing $\sum \sum_{i,j \in[-n,n], (i,j)\ne (0,0)} \frac{(-1)^{j+k}}{\sqrt{j^2+k^2}}$ in a different way

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

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

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 $$\approx 1.164$$.

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 $k=i^2+j^2$

• 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 May 17, 2021 at 12:34
• 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 ? May 17, 2021 at 12:36
• in Math languaje $$\sum _{k=-n}^n \sum _{j=-n}^n \to \sum _{k=1}^n \sum _{j=1}^n$$ May 17, 2021 at 18:22
• You had asked a question previously a question on Madelung constant : you should link your questions in order to help people to help you. May 20, 2021 at 20:53