On comparing the coefficients of general product L-functions

Let f and g be two distinct primitive holomorphic cusp forms of even integral weights \(k_{1}\) and \(k_{2}\) for the full modular group \(\Gamma =SL(2,\mathbb {Z})\), respectively. Denote by \(\lambda _{f\otimes f\otimes \cdots \otimes _{l} f}(n)\) and \(\lambda _{g\otimes g\otimes \cdots \otimes _{l} g}(n)\) 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

$$\begin{aligned} \big \{ p ~ : ~ \lambda _{f\otimes f\otimes \cdots \otimes _{l} f}(p) < \lambda _{g\otimes g\otimes \cdots \otimes _{l} g}(p)\big \}, \end{aligned}$$

where \(l\geqslant 4\) is any fixed integer. By analogy, we also establish some similar density results of the above supported on certain binary quadratic form.