Let \(H_k^*\) be the set of normalized primitive holomorphic cusp forms of even integral weight \(k\) for the full modular group \(\Gamma = \mathrm{SL}(2, \mathbb{Z})\), and denote by \(\lambda_f(n)\) the \(n\)th normalized Fourier coefficient of \(f\in H_k^*\). Let \(\lambda_{f \times f}(n)\) be the \(n\)th normalized coefficient of the Dirichlet expansion of the Rankin-Selberg \(L\)-function \(L(f\times f,s)\) associated with \(f\). The main purpose of this paper is to establish the asymptotic behavior of the sum \[ \sum_{\substack{n \le x\\ n\equiv l\pmod q}} \lambda^2_{ f \times f}(n)=\frac{x}{\varphi(q)}P(\log x)+O_{f, \epsilon}\left(qx^{1-\frac{3}{2}\eta+\epsilon}\right), \] where \(q\) is a prime with \((l, q) = 1\), \(q\ll x^{\eta}\) and \(P(t)\) is a polynomial in \(t\) of degree \(1\) with leading positive coefficient.
In a similar manner, the authors also consider the general divisor problem of the coefficients of Rankin-Selberg \(L\)-functions associated with \(f\) over the set of arithmetic progressions. Actually, they show that \[ \sum_{\substack{n \le x\\ n\equiv l\pmod q}} \lambda_{\omega, f \times f}(n)=\frac{x}{\varphi(q)}P_{\omega-1}(\log x)+O_{f, \epsilon}\left(qx^{1-\frac{3}{2}\vartheta_\omega+\epsilon}\right), \] for any \(\epsilon>0\), \(\vartheta_\omega=\frac{92}{247\omega+10}\), \(q\ll x^{\vartheta_\omega}\), and \(P_{\omega-1}(t)\) is a polynomial in \(t\) of degree \(\omega-1\) with positive leading coefficient.