Hardy-Littlewood Circle Method

Theorem
$\newcommand{\A}{\mathcal A}$Let $\A$ be a subset of the non-negative integers.

Let


 * $\displaystyle T(s) = \sum_{a \in \A} s^a $

be the generating function for $\A$.

For $N \in \N$, let $r_{\A,\ell}(N)$ be the number of solutions $(x_1,\ldots, x_\ell) \in \A^\ell$ to the equation


 * $ x_1 + \cdots + x_\ell = N $

Then


 * $\displaystyle r_{\A,\ell}(N) = \oint_{|s| = \rho} \frac{T(s)}{s^{N+1}}\ ds $

for any $\rho \in (0,1)$.

Proof
We have


 * $\displaystyle T(s)^\ell = \sum_{N = 0}^\infty r_{\A,\ell}(N) s^N $

and


 * $\displaystyle \frac{d^N}{ds^N} \left[ T(s)^\ell \right] = N! \cdot r_{\A,\ell}(N) + O(s) $

so


 * $\displaystyle r_{\A,\ell}(N) = \frac{d^N}{ds^N} \left[ T(s)^\ell \right]_{s = 0} $

Now recall Cauchy's Integral Formula for Derivatives for a complex function $f$ holomorphic on a domain $D$, and a path $\gamma \subseteq D$:


 * $\displaystyle \frac{d^N}{ds^N} f(s)\Big|_{s = a} = \frac{ N! }{2 \pi i} \oint_{\gamma} \frac{f(s)}{\left({ s - a }\right)^{N+1}} \ ds$

Since $T(s)$ is defined by a generating function, $T(s)^\ell$ has a Taylor series about $s = 0$ which converges for all $|s| < 1$. Applying Cauchy's formula we have

where $\gamma$ a circle about zero of radius $\rho$ for any $\rho < 1$.