Preliminaries to the Circle Method and the Method of Modular Functions

We have presented in Chapters 2, 3, and 4 the contributions made by mathematicians, from antiquity until well into the 19th century, to the problem of representations of natural integers by sums of k = 2, 4, and 3 squares. Theorems were obtained that presented the number r k (n) of these representations as sums of divisor functions (for k = 2 and 4) or sums over Jacobi symbols (for k = 3). Next, in Chapters 8 and 9, we studied a method developed mainly by Jacobi, based on the use of theta functions, by which one obtains similar results, involving divisor functions, for an even number of squares. The method is entirely successful for k ≤ 8, while for k ≥ 10 the formulae contain, besides sums of divisor functions, also more complicated additional terms. Even for k ≤ 8, the cases k = 5 and k = 7 were bypassed. Nevertheless, as mentioned earlier (see Chapter 10), Eisenstein presented formulae depending (as in the case k = 3) on Jacobi symbols for r5(n) and formulae for r5(n) of a different type were proposed and proved by Stieltjes (1856–1894) (see [259], [260]), and by Hurwitz [118]. Eisenstein also stated formulae for r7(n), expressed, like those for r5(n) with the help of the Legendre-Jacobi symbol. Proofs of Eisenstein’s formulae were given by Smith [253, 254], who deduced them from his own, of a different appearance, and by Minkowski (1864–1909) (see [178]).