Bounds for Average toward the Resonance Barrier for GL(3) × GL(2) Automorphic Forms.

Let f be a fixed Maass form for SL₃ (ℤ) with Fourier coefficients A_f(m, n). Let g be a Maass cusp form for SL₂ (ℤ) with Laplace eigenvalue 1 4 + k 2 and Fourier coefficient λ_g(n), or a holomorphic cusp form of even weight k. Denote by S_X(f × g, α, β) a smoothly weighted sum of A_f(1, n)λ_g(n)e(αn^β) for X < n < 2X, where α ≠ 0 and β > 0 are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of S_X(f × g, α, β) over g as k tends to ∞ with X. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak. [ABSTRACT FROM AUTHOR]