A general divisor problem involving Hecke eigenvalues

Let f(z) be a holomorphic or Maass cusp form for the full modular group \(SL(2,\mathbb {Z})\) $SL(2,\mathbb{Z})$ . In this paper, we consider the asymptotic behavior of the sums \( U_{w}(x)=\sum _{n\le x}\lambda _{w,f\times f}(n), \) $U_w(x)={\sum }_{n\le x}{\lambda }_{w,f\times f}(n),$ which can be regarded as the general divisor problems considered by Kanemitsu et al. (Monatsh Math 136(1):17–34, 2002), and get some new results which improve previous results.