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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.04409 |
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Table of Contents:
- 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.