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

Question

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

Answer 1

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 $k=i^2+j^2$