Ytukyg

The question

The final goal (for this stage of my project) is to get an explicit form for $phi(n)$. This last one is the number of integer solutions to $x^2+y^2+z^2+2t^2=n$. You may find this $phi(n)$ on OEIS or you can find the first several numbers by pasting the following code into something like mathematica:

CoefficientList[(1 + 2 Sum[q^(2(j)^2), {j, 10}] ) *(1 + 2 Sum[q^((j)^2), {j, 10}] )^3 , q]

The question is what's an explicit form for $phi(n)?$

Exposition

I would like to get an explicit formula for $phi(n)$. Let me tell you what I know so far. It looks like $phi(n)$ is well-behaved on odd $n$ and on powers of $2$.

$chi(x)= sqrt{2}sin(frac{pi}{4}x+pi)=begin{cases} 1 hspace{1 cm} text{when }x equiv 1,7 mod 8
\ -1 hspace{1 cm} text{when }x equiv 3,5 mod 8 end{cases}$

For $nequiv 1, 7 mod 8$ we have

$$phi(n)=6sum_{d|n} {chi(d)}d$$

For example, $33$ is congruent to $1 mod 8$ and indeed $phi(33)=6times(1-3-11+33)=120$

For $n equiv 3,5 mod 8$ we have

$$phi(n)=-10sum_{d|n} {chi(d)}d$$

For powers of $2$ we have $phi(2^alpha)=2^{alpha+3}-2$

I am using as a type of template here Joseph Liouville's Sur La Forme $x^2+y^2+z^2+3t^2$ and Sur La Forme $x^2+y^2+z^2+5t^2$. I am not able to complete the characterization and I am not $100 %$ sure that Liouville does this for the forms above either(though I suspect he does) because I don't speak/read French very well (though math is math and this I can read). I can't find anything Liouville wrote on $x^2+y^2+z^2+2t^2$. If anyone knows that he surely did and can point me to the right spot I would appreciate it. Also if this is in Grosswald's text (which I don't own) I would definitely reconsider purchasing it. It's not clear to me from the TOC whether this is the right place to look. Also (and I sincerely doubt this one)... you know if Liouville is one of these mathematicians where everything has already been translated and I can find it English that would be amazing.

How do I finish the job and complete this characterization?

Motivations!
What I think would be very cool would be to argue that $ sum_{n=1}^R phi(n)$ is approximately the interior volume of $x^2+y^2+z^2+2t^2=R$ and thereby win a series of rational numbers that converges to the volume in the interior of $x^2+y^2+z^2+2t^2=1$ (which I would guess should be $frac{pi^2}{2sqrt{2}}$? Correct me if I am wrong but this is kind of a minor detail.) I have explained that technique (perhaps ad nauseam) over here.

Letting $phi(n)$ denote the number of integer solutions to $x^2+y^2+z^2+2t^2$, with $n=2^{alpha}N$ where $N$ is odd we have
$$phi(n)=2bigg( 2^{alpha+2}bigg( frac{ 8}{N} bigg)-1 bigg) sum_{d|N} bigg( frac{ 8}{d} bigg)d$$

Where above appears Legendre-Jacobi-Kronecker symbols. In particular,

$$bigg( frac{8}{x} bigg) = cases{ hspace{0.32 cm} 0 text{ if } x equiv 0 hspace{.44 cm} (mod 2) \hspace{0.32 cm} 1 text{ if } x equiv 1,7 (mod 8) \ -1 text{ if } x equiv 3,5 (mod 8)}$$

An elementary proof can be found here.

So the next step in this process is to look for the asymptotic behavior of $sum_{n=1}^x phi(n)$ and see whether we can apply something like Abel's Summation here.