Luo's Spectral Large Sieve Inequality on Short Intervals
Zhi Qi, Ruihua Qiao
Abstract
Let uj traverse an orthonormal basis of Hecke--Maass forms for SL2 ( 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 λj (n) nitj on the range tj ≤slant T and prove that the `Eisenstein--Kloosterman' cancellation discovered by Luo is effective on the interval T < tj ≤slant T + M as long as T < M ≤slant T. Moreover, our approach yields an improvement of the large sieve inequality of Luo.
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