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This is a question that i have already asked on HSM stackexchange, and i decided to ask it again here because it's more mathematical than historic (to make a conclusion in this question one needs more mathematical then historical understanding). 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 identities on the 4th power of the theta function:

$(\sum_{-\infty}^\infty q^{{n^2}})^4 = (\sum_{-\infty}^\infty (-1)^n q^{{n^2}})^4 + (\sum_{-\infty}^\infty q^{{(2n - 1)^2/4}})^4 = 1 + 8\sum_{1\le m} \frac {{mq^m}}{{1 - (-1)^{m + 1}q^m}} = 1 + \sum_{1 \le m}\hat \sigma (k)q^k$

The point is that the last equality means that the coefficients of the $k$th power in the right side of the last equallity must be equal to $r_4(k)$ (number of representations of $k$ as sum of $4$ squares), and an additonal interpretation (by certain manipulations) of the right side of the equallity 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 dont understand the subject enough to make a conclusion. So i'll be glad to hear an expert 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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  • $\begingroup$ My question is very historical in nature but requires deep mathematical understanding in order to answer it (i'm doing a historical project on C.F Gauss). In other words, my question can be reformulated as: does the derivation of the identities of Gauss imply that Gauss possesed the critical techniques for proving the four squares theorem? and in addition was Gauss aware of the interpretation of his results on theta functions as being related to the sum of squares function? $\endgroup$ – user2554 Dec 22 '17 at 12:52
  • $\begingroup$ Thus to answer we need to write down the sketch proof and techniques. Do you have some of them ? And it is clear Gauss knew $(\sum_n q^{n^2})^2 =1+4 \sum_k q^k \sum_{d | k} \chi_4(d)$ (Fermat two square theorem) $\endgroup$ – reuns Dec 22 '17 at 14:04
  • $\begingroup$ I know how to deduce from Gauss's identity: $(\sum_{-\infty}^\infty q^{{n^2}})^4 = 1 + 8\sum_{1\le m} \frac {{mq^m}}{{1 - (-1)^{m + 1}q^m}}$ the specific form of Jacobi's theorem - $r_4(k) = 8\sigma(k)$ or $24\sigma(k)$, depends if k is odd or even (by using elementary number theoretic demonstrations) . But i have no idea how to derive this identity; according to what i've read, it's derived from the advanced mathematical theories of elliptic functions, theta functions and modular forms. $\endgroup$ – user2554 Dec 22 '17 at 14:25
  • $\begingroup$ To make the deduction, one simply needs to notice that the sum is actually a double sum $\sum_{n = 1}^\infty n (\sum _{k = 1}^\infty (-1)^{(k+1)(n+1)}q^{kn})$ , a result of expansion in a geometric series. $\endgroup$ – user2554 Dec 22 '17 at 14:31
  • $\begingroup$ For the method using that both sides are modular forms $\in M_2(\Gamma_0(4))$, see p.33 here $\endgroup$ – reuns Dec 22 '17 at 14:34
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Gauss knew a whole lot about theta functions and their modular properties. But his ideas regarding these functions were published posthumously. You can find a detailed description of the development done by Gauss in the famous paper of David Cox titled AGM of Gauss.

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