The spectral mean value for linear forms in twisted coefficients of cusp forms. (English) Zbl 0821.11035

This paper concerns the large sieve type inequality for the twisted Fourier coefficients of Maass cusp forms. The approach goes through the complete spectral resolution of the Laplacian via Kuznetsov’s formula. The desired inequality for the contribution from the discrete spectrum is achieved after the dominating term from the continuous spectrum cancels out with a part of sum of the Kloosterman sums. The result has application to estimating the size of the automorphic \(L\)-functions at the special points.
Reviewer: W.Luo (Berkeley)


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11N36 Applications of sieve methods
11L05 Gauss and Kloosterman sums; generalizations
11F30 Fourier coefficients of automorphic forms
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