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arXiv · 2608.29558

Luo's Spectral Large Sieve Inequality on Short Intervals

Zhi Qi·Ruihua Qiao
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Abstract

Let $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues $\lambda_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ \lambda_j (n) n^{it_j} $ on the range $t_j \leqslant T$ and prove that the `Eisenstein--Kloosterman' cancellation discovered by Luo is effective on the interval $ T < t_j \leqslant T + M $ as long as $\sqrt{T} < M \leqslant T$. Moreover, our approach yields an improvement of the large sieve inequality of Luo.

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Zhi Qi, Ruihua Qiao. 2026-08-30. Luo's Spectral Large Sieve Inequality on Short Intervals. https://arxiv.org/abs/2608.29558

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