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In p. 283-285 of volume 2 of Dickson's “history of the theory of numbers” appear several formulas of striking similarity: some of them are stated by Gauss (p.283) and some are stated by Jacobi (p.285); they are actually the same and only the notation differs ($y$ in Gauss's formula and $q$ in Jacobi's formula). Gauss's formulas are the following:

$(1 + 2y + 2y^4 + 2y^9 + ... + 2y^{{n^2}})^4 = (1 - 2y + 2y^4 - ...)^4 + (2y^{{1/4}} + 2y^{{9/4}} + 2y^{{(2n - 1)^2/4}})^4$

and

$(1 + 2y + 2y^4 + 2y^9 +... + 2y^{{n^2}})^4 = 1 + 8(y/(1 - y) + 2y^2/(1 + y^2) + 3y^3/(1 - y^3) + .. + ny^n/(1 + (-1)^n y^n))$

The point is that the last equality means that the coefficients of the $k$th power in the right side of the last equality must be equal to $r_4(k)$ (number of representations of $k$ as sum of $4$ squares), and an additional interpretation (by certain manipulations) of the right side of the equality gives the result of Jacobi: $r_4(k) = 8\sigma(k)$ or $24\sigma(k)$, depends if $k$ is odd or even.

Since at the same posthumous paper Gauss makes some remarks on the representation of numbers as sum of four squares, it seems to me very probable that Gauss did make the deduction of the four squares theorem about the number of representations as sum of four squares from these identities.

I ask this question because I don't understand the subject enough to make a conclusion. So I'll be glad to hear an expert's opinion on this matter.

Update: For the completeness of the discussion i must add another relevant reference. In the chapter "sum of four squares" of volume 2 of Dickson's work, at page 300, he mentions that the czech mathematician Karel Petr proved two formulas by Gauss (Werke, III,p. 476) on theta functions by the method outlined by Gauss. The point is that K. petr used those identities of Gauss to derive relations giving the number of representations of a number N by three quaternary quadratic forms: $x^2 + y^2 + 9z^2 + 9u^2$, $x^2+y^2+z^2+9u^2$, $x^2+9y^2+9z^2+9u^2$.

I mention this fact (which i noticed only in the last days) because it seems now that Gauss-Jacobi identity wasn't an isolated result, but was part of a grand plan Gauss had for the subject of analysis, and theta functions, in particular.

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  • $\begingroup$ Your question would be more self-contained if you include some of the formulas you mention here only by page number. Also, the first sentence is not really necessary. $\endgroup$ – KCd Jul 5 '17 at 13:26
  • $\begingroup$ i've included the formulas. $\endgroup$ – user2554 Jul 5 '17 at 14:51

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