Gravitating lattice Fourier kernel
= Gravitating lattice Fourier kernel
{title2=$F(\xi)=2\sum_{l\ge1}(1-\cos(l\xi))/l^3$}
The discrete normal modes of an <infinite gravitating particle lattice> involve $F(\xi)=2\sum_{l=1}^\infty(1-\cos(l\xi))/l^3$. It is even, nonnegative and $2\pi$-periodic. Its derivative is a positive reciprocal-square sine series for $0<\xi<\pi$, so its maxima occur exactly at odd multiples of $\pi$, with $F(\pi)=(7/2)\zeta(3)$.