Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 321 / 3 / c / Solution

Set $\xi =hk$. For the discrete Fourier mode, the displacement difference in the $l$th neighbor pair carries $e^{il\xi }-1$, andThus the infinite system reduces toThe gravitating lattice Fourier kernel $F$ is nonnegative, even and $2\pi$-periodic, with $F(0)=0$. The uniform convergence and absolute convergence of its differentiated series givesTo prove the maximum, it is not sufficient to maximize each cosine term separately: the even terms do not reach their individual maxima at $\pi$. Instead use the convergent integral representation $l^{-2}={\int }_0^{\infty }t{e}^{-lt}dt$. Summing the geometric series for sines givesFor $0<\xi <\pi$, the denominator is positive and $\sin \xi >0$, so $F^{\prime }>0$. For $\pi <\xi <2\pi$, $F^{\prime }<0$. The interchange is justified by ${\sum }_l{\int }_0^{\infty }t{e}^{-lt}|\sin (l\xi )|dt\le {\sum }_l{l}^{-2}<\infty$. HenceTheir value iswhere $\zeta$ is the Riemann zeta function. This maximizing Fourier mode alternates the signs of neighboring particle displacements.