On comparing the coefficients of general product L-functions

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1}\) $k_1$ and \(k_{2}\) $k_2$ for the full modular group \(\Gamma =SL(2,\mathbb {Z})\) $\mathrm{\Gamma }=SL(2,\mathbb{Z})$ , respectively. Denote by \(\lambda _{f\otimes f\otimes \cdots \otimes _{l} f}(n)\) $\lambda _{f\otimes f\otimes \cdots {\otimes }_{l}f}^{}(n)$ and \(\lambda _{g\otimes g\otimes \cdots \otimes _{l} g}(n)\) $\lambda _{g\otimes g\otimes \cdots {\otimes }_{l}g}^{}(n)$ the nth normalized coefficients of the l-fold product product L-functions attached to f and g, respectively. In this paper, we establish a lower bound for the analytic density of the set \(\begin{aligned} \big \{ p ~ : ~ \lambda _{f\otimes f\otimes \cdots \otimes _{l} f}(p) < \lambda _{g\otimes g\otimes \cdots \otimes _{l} g}(p)\big \}, \end{aligned}\) $$\begin{array}{c}\hfill \{p:\lambda _{f\otimes f\otimes \cdots {\otimes }_{l}f}^{}(p)<\lambda _{g\otimes g\otimes \cdots {\otimes }_{l}g}^{}(p)\},\end{array}$$ where \(l\geqslant 4\) $l⩾4$ is any fixed integer. By analogy, we also establish some similar density results of the above supported on certain binary quadratic form.