The average behaviour of Fourier coefficients of the Hecke–Maass form associated to k-free numbers

Let f and g be two distinct normalized primitive Hecke–Maass cusp forms of weight zero with Laplacian eigenvalues \(\frac{1}{4}+u^{2}\) and \(\frac{1}{4}+v^{2}\) for the full modular group \(\Gamma =SL(2,\mathbb {Z})\), respectively. Denote by \(\lambda _{f}(n)\) and \(\lambda _{g}(n)\) the nth normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the non-trivial upper bounds for the sum \(\sum _{n\in S}|\lambda _{f}(n)\lambda _{g}(n)|\), where S is a suitable subset of \(\mathbb {Z}^{+}\cap [1,x]\) with certain properties.