Discrete mean square of the coefficients of triple product \(L\)-functions over certain sparse sequence (Q2088655)

Let \(f\), \(g\) and \(h\) be three distinct normalized primitive holomorphic cusp forms of even integral weights \(k_1\), \(k_2\) and \(k_3\) for the full modular group \(\mathrm{SL}(2, \mathbb{Z})\), respectively. Let \(\lambda_{f\times g\times h}(n)\) denote the \(n\)-th coefficient of triple product \(L\)-function \(L(f \times g \times h, s)\) associated to \(f\), \(g\), \(h\). The present papers gives the average behavior of the following sum: \[ \sum_{\substack{n=a_1^2+a_2^2+a_3^2+a_4^2\leq x\\ (a_1,a_2, a_3, a_4)\in \mathbb{Z}^4}}\lambda^2_{f\times g\times h}(n)=c x^2+ O_{\varepsilon}(x^{65/33+\varepsilon}), \] where \(c\) is an effective constant depending on \(f\), \(g\) and \(h\) and \(\varepsilon\) is an arbitrarily small positive constant.