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 ${\lambda }_j(n)$ 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 ${\lambda }_j(n)n^{i{t}_j}$ on the range $t_j⩽T$ and prove that the `Eisenstein--Kloosterman' cancellation discovered by Luo is effective on the interval $T<t_j⩽T+M$ as long as $\sqrt{T}<M⩽T$. Moreover, our approach yields an improvement of the large sieve inequality of Luo.

17 pages; results improved; to appear in Sci. Sin. Math. (in Chinese)