Does the following double sum converge and if so, do we have a closed form ?
$$ A = \zeta_{17}(1)= \lim_{s=1} \lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k}^{++} \frac{1}{(a^2 + 17 b^2)^s} - \sum_{k<a^2 + 17 b^2 < k^2}^{++} \frac{1}{(a^2 + 17 b^2)^s}$$
Where the $++$ indicates we only take nonnegative $a,b$.
Motivation
Notice this is - or is suppose to be - some kind of regularized zeta function.
It is an analogue of some regularized Riemann zeta function with $\zeta(1) = \gamma$ and agreeing with the usual Riemann zeta for $1< Re(s)$. Basically we "removed" the pole (with residue $1$) at $\zeta(1)$.
I got inspired by quadratic rings and zeta functions. I know for instance that the norm $a^2 + 17 b^2$ for the ring $\Bbb Z[\sqrt{-17}]$ is multiplicative yet the ring $\Bbb Z[\sqrt{-17}]$ is not a UFD !!
I also more or less know norms are used in generalized zeta functions.
This probably relates to elliptic functions and double periodic functions and such, but I am not confident about that.
We could drop the $s$ completely actually and write :
$$ A = \lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k}^{++} \frac{1}{a^2 + 17 b^2} - \sum_{k<a^2 + 17 b^2 < k^2}^{++} \frac{1}{a^2 + 17 b^2}$$
Where the $++$ indicates we only take nonnegative $a,b$.
Why could it have a closed form ?
A few years ago I noticed
$$ \sum_{m,n = - \infty}^{\infty} \frac{(-1)^m}{m^2 + 58 n^2} = - \frac{\pi \ln( 27 + 5 \sqrt {29})}{\sqrt {58}} $$
Where the sum omits the case $n = m = 0$ ofcourse.
And posted it as a question here ,
The situation here is maybe a bit different but maybe $A$ can be expressed by
$B = \lim_{k \to \infty} \sum_{0<a^2 + 17 b^2 < k}^{++} \frac{(-1)^a}{a^2 + 17 b^2}$
and then by using Dirichlet-L functions, zeta functions, double periodic or q-series maybe a closed form.
note : I left out the physics perspective.
Numerical values are also appreciated !
edit
A guess
We know the analogue case with the Riemann zeta and euler constant can be given by taking and comparing the integral and sum of $1/x$.
Inspired by that I considered a guess based on
$$\int \int \frac{1}{x^2 + 17 y^2}$$
and ignoring constants of integration and such. So basicly terrible math and an "euler proof" at best if it is correct. My apologies.
But maybe it is correct and can lead to an actual proof.
Anyway inspired by the integral the guess is :
$$A = \frac{Im[Li_2(\sqrt{-17})]}{\sqrt{17}}$$
where $Im$ is the imaginary part and $Li_2(x)$ is a polylogarithm.
A weaker conjecture is that that value is the tail of the limiting sum $A$ and hence the actual value is the above + a rational number based on the few first terms of the sum in $\zeta_{17}$.
The integral idea is based on truncating $a$ and $b$ rather then $a^2 + 17 b^2$ but assuming it should give the same result. That is not rigor ofcourse, I am aware.
Just my 2 cent.
Like I said, numerical results are welcome and they might conclude if this conjecture makes sense or not.
It might be possible the polylog is involved but it requires some adjustements.
I have additional questions but I will post them seperately.
