A spectral resolution of the large sieve

The quadratic form $V(\phi ,Q)={\sum }_{q\sim Q}{\sum }_{a\mathrm{m}\mathrm{o}\mathrm{d}{}^{\ast }q}\mathrm{\mid }S(\phi ,a\mathrm{/}q){\mathrm{\mid }}^2$ and its eigenvalues are well understood when $Q=o(\sqrt{N})$, while $V(\phi ,Q)$ is expected to behave like a Riemann sum when $N=o(Q)$. The behavior in the range $Q\in [\sqrt{N},100N]$ is still mysterious. In the present work we present a full spectral analysis when $Q\ge N^{7\mathrm{/}8}$ in terms of the eigenvalues of a one-parameter family of nuclear difference operators. We show in particular that (a smoothed version of) the quadratic form $V(\phi ,Q)$ may stay \emph{away} from $(6\mathrm{/}{\pi }^2)Q{\sum }_n\mathrm{\mid }{\phi }_n{\mathrm{\mid }}^2$ when $Q\asymp N$, though only on a vector space of positive but small dimension. An improved version of this paper, with the same title, will appear (2024 or 2025) in the Bulletin of the French Mathematical Society.