TITLE: Did Gauss know Jacobi's four squares theorem?
QUESTION [5 upvotes]: 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 ($x$ 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.
In the same passage from Gauss's nachlass (Werke, volume 3, p. 444-445, passage [9])  in which he writes down Jacobi's identity, and just before this identity, Gauss also writes down $\mathbb{log}(\vartheta_3^4(x))$ as:
$$\mathbb{log}((1+2x+2x^4+2x^9+\cdots)^4) = 8\cdot (\frac{x}{1+x}+\frac{x^3}{3(1+x^3)}+\frac{x^5}{5(1+x^5)}+\cdots)$$,
(actually he writes down the series for $\frac{1}{2}\mathbb{log}(\vartheta_3(x))$, but it is equivalent to what i wrote). Immediately after writing down several identities on the fourth powers of Jacobi theta functions $p,q,r$, Gauss proceeds and writes a differential equation satisfied by new variables $t,u$ (defined by : $t = \frac{1}{p^2}, u  = \frac{1}{q^2}$)  and their first, second and third derivatives (of $t,u$).
Since i'm unfamiliar with the theory of modular forms, i'm unable to see how Gauss arrived at this identity for $\mathbb{log}(\vartheta_3^4(x))$, nor i'm able to see how one can find the series developement of $\vartheta_3^4(x)$ from that of $\mathbb{log}(\vartheta_3^4(x))$. But maybe some of the mathematicians here who are familiar with modular forms can see the connection.

REPLY [4 votes]: 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.