The Circle Method

We mentioned in Chapter 1 that the number r _s (n) of solutions of the Diophantine equation (12.1)$$ \sum\limits_{{k = 1}}^s {x_i^2} = n $$ is the coefficient of xn in the Taylor expansion of the function $$ 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_s}(n){x^n}} $$. Here, as in Chapter 8, we write θ(x) for θ₃(1;x) and we shall suppress the first entry, which will always be z = 1. From (12.1); it follows, by Cauchy’s theorem, that (12.2)$$ {r_s}(n) = \frac{1}{{2\pi i}}\int\limits_c {{x^{{ - n - 1}}}{\theta^s}(x)dx,} $$ where, we recall, (12.3)$$ \theta (x) = \sum\limits_{{ - \infty }}^{\infty } {{x^{{{n^2}}}}} = 1 + 2\sum\limits_{{n = 1}}^{\infty } {{x^{{{n^2}}}}} = \sum\limits_{{k = 0}}^{\infty } {{a_k}{x^k},\quad say,} $$ and ?? is a sufficiently small circle around the origin.