On the binary additive divisor problem in mean

We study a mean value of the classical additive divisor problem, that is∑f∼F∑n∼N|∑l∼Ld(n+l)d(n+l+f)−main term|2, with quantities N≥1, 1≤F≪N1−ε and 1≤L≤N. The main term we are interested in here is the one by Motohashi [27], but we also give an upper bound for the case where the main term is that of Atkinson [1]. Furthermore, we point out that the proof yields an analogous upper bound for a shifted convolution sum over Fourier coefficients of a fixed holomorphic cusp form in mean.