The Circle Method

Emil Grosswald
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Emil Grosswald: Temple University, College of Liberal Arts

Chapter Chapter 12 in Representations of Integers as Sums of Squares, 1985, pp 149-174 from Springer

Abstract: Abstract 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 x n 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 𝓒 is a sufficiently small circle around the origin.

Date: 1985
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DOI: 10.1007/978-1-4613-8566-0_13

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