PreprintCC BY 4.0arXiv
Luo's Spectral Large Sieve Inequality on Short Intervals
Abstract
Let $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues λj (n) and Laplace eigenvalue 1/4+tj2. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ λ_j (n) n^{it_j} $ on the range tj 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 √T < M leqslant T. Moreover, our approach yields an improvement of the large sieve inequality of Luo.
Related
- The large sieve inequality and additive decompositions of sums of squares
- Large value estimates for Dirichlet polynomials and trilinear sums over short intervals
- An improvement of the large sieve for square moduli under Hooley's hypothesis R* for short Salié sums
- The weighted large sieve through Parseval
- The large sieve through Parseval, Large sifted sets are regular
BibTeX
@misc{qi2026luos,
title = {{Luo's Spectral Large Sieve Inequality on Short Intervals}},
author = {Zhi Qi and Ruihua Qiao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.29558},
eprint = {2608.29558},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29558},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Luo's Spectral Large Sieve Inequality on Short Intervals
AU - Zhi Qi
AU - Ruihua Qiao
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.29558
UR - https://arxiv.org/abs/2608.29558
AB - Let $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues λ_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 $ λ_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 √T < M leqslant T. Moreover, our approach yields an improvement of the large sieve inequality of Luo.
ER -