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 \(\frac{1}{4}+u^{2}\) $\frac{1}{4}+u^2$ and \(\frac{1}{4}+v^{2}\) $\frac{1}{4}+v^2$ for the full modular group \(\Gamma =SL(2,\mathbb {Z})\) $\mathrm{\Gamma }=SL(2,\mathbb{Z})$ , respectively. Denote by \(\lambda _{f}(n)\) ${\lambda }_f(n)$ and \(\lambda _{g}(n)\) ${\lambda }_g(n)$ the nth normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the non-trivial upper bounds for the sum \(\sum _{n\in S}|\lambda _{f}(n)\lambda _{g}(n)|\) ${\sum }_{n\in S}|{\lambda }_f(n){\lambda }_g(n)|$ , where S is a suitable subset of \(\mathbb {Z}^{+}\cap [1,x]\) ${\mathbb{Z}}^{+}\cap [1,x]$ with certain properties.