Year: 2025
Type: article
Abstract: The quadratic form $V(φ,Q)=\sum_{q\sim Q}\sum_{a\mod^* q}|S(φ,a/q)|^2$ and its eigenvalues are well understood when $Q=o(\sqrt{N})$, while $V(φ,Q)$ is expected to behave like a Riemann sum when $N=o(Q)$. The behavior in the range $Q\in[\sqrt{N},100 N]$ is still mysterious. In the present work we present a full spectral analysis when $Q\ge N^{7/8}$ 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 $V(φ,Q)$ may stay \emph{away} from $(6/π^2)Q\sum_n|φ_n|^2$ 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.
DOI: 10.24033/bsmf.2907
Author: Olivier RAMARÉ
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4323634079