THE THREE-DIMENSIONAL DIVISOR PROBLEMS RELATED TO CUSP FORM COEFFICIENTS

Abstract

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\mathrm{\Gamma }=\text{SL}(2,\mathbb{Z})$. Let ${\lambda }_{f\times f}(n)$ and ${\lambda }_{f\times f\times f}(n)$ be the normalized coefficients of the Dirichlet expansion of the Rankin–Selberg $L$-function and triple product $L$-function attached to $f$, respectively. We establish the asymptotic formulae and the upper bounds for the three-dimensional divisor problems related to these normalized coefficients, respectively.