Resonance sums for Rankin–Selberg products of SLm(Z) Maass cusp forms

a r t i c l e i n f o a b s t r a c t Let f and g be Maass cusp forms for SL m (Z) and SL m (Z), respectively, with 2 ≤ m ≤ m. Let λ f ×g (n) be the normalized coefficients of L(s, f × g), the Rankin–Selberg L-function for f and g. In this paper the asymptotics of a Voronoi-type summation formula for λ f ×g (n) are derived. As an application estimates are obtained for the smoothly weighted average of λ f ×g (n) against e(αn β). When β = 1 mm and α is close or equal to ±mm q 1 mm for a positive integer q, the average has a main term of size |λ˜f טg (q)|X 1 2mm + 1 2. Otherwise, when 0 < β < 1 mm , it is shown that this average decays rapidly. This phenomenon is due to the oscillatory nature of the coefficients λ f ×g (n).