Worked solution: Viazovska's modular form solution to sphere packing in dimensions 8 and 24

The Poisson summation formula relates sums of a function over a lattice to sums of its Fourier transform over the dual lattice; for the $E_8$-lattice packing $\tfrac{1}{\sqrt2}\Lambda_8$ this gives $\sum_{\ell\in\frac{1}{\sqrt2}\Lambda_8} f(\ell) = 2^4 \sum_{\ell\in\sqrt2\Lambda_8} \hat f(\ell)$. Combined with the sign conditions $f\le 0$ outside the ball and $\hat f \ge 0$ everywhere, the left side is at most $f(0)$ (only the origin term can be positive) and the right side is at least $2^4 \hat f(0)$ (only the origin term is guaranteed, and every other term is $\ge 0$), which forces $f(0)/\hat f(0) \ge 2^4$, exactly reproducing the Cohn–Elkies bound for the $E_8$ density.

Equality throughout this chain — the only way to actually prove $E_8$ is optimal rather than merely bound it — requires $f(\ell)=0$ for every nonzero $\ell \in \tfrac{1}{\sqrt2}\Lambda_8$ (every other packing vector) and $\hat f(\ell) = 0$ for every nonzero $\ell \in \sqrt2\Lambda_8$. Since $E_8$ vectors come in shells of length $\sqrt{2n}$, this means $f$ and $\hat f$ must each vanish, to a very specific double-zero order, at every one of these shell radii; constructing such a pair is exactly the analytic problem solved in the next two steps.