On sums of Hecke series in short itervals

2003/11/21 by Aleksandar Ivić · 1 citation
Mathematics · #math.NT #msc:11F72 #msc:11F66 #msc:11M41 #msc:11M06

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published as Journal de Theéorie des Nombres de Bordeaux 13(2001), 453-468 · 16 pages

arxiv created 2003/11/21 · arxiv updated 2009/12/01

Abstract

We prove ∑_K-G≤ κ_j ≤ K+Gα_j H_j³(1/2) ≪_εGK^1+ε for K^ε≤ G ≤ K, where α_j = |ρ_j(1)|²(\cosh πκ_j)^-1, and ρ_j(1) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ_j = κ_j² + 1/4 to which the Hecke series H_j(s) is attached. This result yields the new bound H_j(1/2) ≪_εκ_j^1/3+ε.

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