On the shifted convolution problem in mean

We study the following mean value of the shifted convolution problem:∑f∼F∑n∼N|∑l∼Lt(n+l)t(n+l+f)|2, over the Hecke eigenvalues of a fixed non-holomorphic cusp form with quantities N⩾1N⩾1, 1⩽L⩽N1−ε1⩽L⩽N1−ε and 1⩽F≪N2/51⩽F≪N2/5. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds for the shifted convolution problem over the Fourier coefficients of a fixed holomorphic cusp form in mean.