I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the integrals $\int_{2}^{X} \Delta^2(x) \Delta(\alpha x) dx$ and $\int_{2}^{X} E^2(x) E(\alpha x)dx,$ where $\Delta(x)$ is the remainder term in Dirichlet Divisor problem and $E(x)$ is the remainder term of the mean-square formula of the Riemann zeta-function, to show that for a specific type of real number $\alpha,$ the main terms of these two integrals are of the same magnitude, but opposite in signs and thus he proved an intrinsic difference between $\Delta(x)$ and $E(x)$ which has not appeared in the literature. I want to use the Kong Kar Lun's method, I mean the truncated Voronoi formula, to compute the integral$\int_{2}^{X} A^2(t) A(\alpha t)dt.$ So, I need to know the motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt,$ where $\alpha$ is a positive real belonging to $Q[\sqrt{N}]$ and $N$ is a square-free integer and $A(x)$ is the sum of Hecke eigenvalues.
If $f$ is a primitive form of an even weight $k$ for the full modular group $SL_2(Z),$ the function $A$ is defined by $A(x)=\sum_{n<=n} \lambda_f(n),$ where $\lambda_f(n)=\hat f(n)/n^{(k-1)/2}.$
Sincerely,
Khadija.