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 $\phi(n)?$


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 $n\equiv 1, 7 \mod 8$ we have

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

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

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

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

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 $\frac{\pi^2}{2\sqrt{2}}$? Correct me if I am wrong but this is kind of a minor detail.) I have explained that technique (perhaps ad nauseam) over here.

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    $\begingroup$ @dmtri. I prefer something with mathematica over like mathematica. Can you explain why "like" is superior? Maybe: "by running the following code in mathematica:" is better? Otherwise, I really appreciate the edit. Thank you. $\endgroup$ – Mason Dec 29 '18 at 14:29
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    $\begingroup$ Ken Williams, Number Theory in the Spirit of Liouville, cambridge.org/core/books/… has a chapter on your equation. $\endgroup$ – Gerry Myerson Dec 29 '18 at 14:40
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    $\begingroup$ @GerryMyerson. Yay! Thanks so much. $\endgroup$ – Mason Dec 29 '18 at 14:41
  • $\begingroup$ @Mason, you are right, I misunderstand your phrase.:) $\endgroup$ – dmtri Dec 29 '18 at 15:16
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    $\begingroup$ @GerryMyerson. Absolutely. I am trying my best to wiggle through the paywalls. I am cheap/poor so if I can get this information by visiting UMD which isn't too far and not paying $60 I will but anyway: I appreciate the lead. I probably won't get back to this project for another couple weeks. Winter Break allows for some play time but this is not a small project for me and I want to do good time-budgeting. $\endgroup$ – Mason Dec 30 '18 at 3:32

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 $$\phi(n)=2\bigg( 2^{\alpha+2}\bigg( \frac{ 8}{N} \bigg)-1 \bigg) \sum_{d|N} \bigg( \frac{ 8}{d} \bigg)d$$

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.


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