On the sign changes of the coefficients attached to triple product \(L\)-functions (Q6591607)

Let \(f\) and \(g\) be two distinct normalized primitive holomorphic cusp forms of even integral weights \(k_1\) and \(k_2\) for the full modular group \(\Gamma=\mathrm{SL}(2, \mathbb{Z})\). Let \(\lambda_{f \times f \times f}(n)\), and \(\lambda_{f \times f \times g}(n)\) denote the \(n\)th coefficients of the triple product \(L\)-functions \(L(f\times f\times f,s),L(f\times f\times g,s)\) attached to \(f\) and \(f, g\), respectively. In this paper, it is shown the following:\N\begin{itemize}\N\item[(i)] For \(\delta>\frac{2903}{3008}\), the sequence \(\{ \lambda_{f \times f \times f} ( n ) \}_{n \in \mathbb{N}}\) changes its signs at least \(\gg x^{1-\delta}\) times in the interval \((x, 2x]\) for sufficiently large \(x\). In particular, the sequence \(\{ \lambda_{f \times f \times f} ( n ) \}_{n \in \mathbb{N}}\) has infinitely many sign changes.\N\N\item[ (ii)] For \(\eta>\frac{1223}{1265}\), the sequence \(\{ \lambda_{f \times f \times g} ( n ) \}_{n \in \mathbb{N}}\) changes its signs at least \(\gg x^{1-\eta}\) times in the interval \((x, 2x]\) for sufficiently large \(x\). In particular, the sequence \(\{ \lambda_{f \times f \times g} ( n ) \}_{n \in \mathbb{N}}\) has infinitely many sign changes.\N\end{itemize}