Resonances and Ω-results for exponential sums related to Maass forms for SL(n,Z)

Abstract

We obtain resonances for short exponential sums weighted by Fourier coefficients of Maass forms for SL(n, Z). This involves deriving asymptotics for the integrals appearing in the GL(n) Voronoi summation formula. As an application, we also prove an Ω-result for short sums of Fourier coefficients. ‡ Correspondence by email to esa.vesalainen@helsinki.fi, by phone to +358 (0) 44 562 5504, or by mail to Esa V. Vesalainen, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, FINLAND. Iwaniec, Luo and Sarnak [16] and by Ren and Ye [31].Ω-results are the other side of the story. With an Ω-result we mean a lower bound in the following sense: M m M +∆ a(m) = Ω(F (M, ∆)) means that M m M +∆ a(m) = o(F (M, ∆)). The result above (2) has been used to prove the following Ω-result: M m M +cM 1/2 a(m) = Ω(M 1/4 ),

Computing the Laplace eigenvalue and level of Maass cusp forms

“…Ren and Ye then gave resonance results for SL (3, Z) Maass cusp forms in [18] and [20]. Next, Ren and Ye in [19] and Ernvall-Hytönen-Jääsaari-Vesalainen in [7] considered resonance for SL(n, Z) Maass cusp forms for n ≥ 2. Finally, resonance sums were considered in special cases such as Rankin-Selberg products in [6], arithemetic functions relating to primes in [23], and used to derive bounds in terms of the spectral parameter r in [22].…”

On Short Sums Involving Fourier Coefficients of Maass Forms

“…An elementary application of Stirling's formula says that when s lies in the vertical strips −δ ≤ (s) ≤ 1 + δ, for a small fixed δ > 0, and has sufficiently large imaginary part, the multiple Γ-factors can be replaced by a single quotient of two Γ-factors [4]:…”

On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$

“…This produces an L-function attached to the form f called the Godement-Jacquet L-function. An elementary application of Stirling's formula says, that when s lies in the vertical strips −δ ≤ ℜ(s) ≤ 1 + δ, for a small fixed δ > 0, and has sufficiently large imaginary part, the multiple Γ-factors can be replaced by a single quotient of two Γ-factors [4]:…”