Deterministic comparison of cusp form coefficients over certain sequences

Abstract

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 $\lambda _{f}(n)$ and $\lambda _{g}(n)$ the nth normalized Fourier coefficients of f and g, respectively. And set $Q(\textbf{x})$ a primitive integral positive-definite binary quadratic form of fixed discriminant $D<0$ with the class number $h(D)=1$. In this paper, we establish a lower bound for the analytic density of the set

$$\begin{aligned} \big \{ p ~: ~ p=Q(\textbf{x}) ~ \text { for some }~ \textbf{x}\in {\mathbb {Z}}^{2}, ~\lambda _{f}(p^{i})\lambda _{f}(p^{j}) < \lambda _{g}(p^{i})\lambda _{g}(p^{j})\big \}, \end{aligned}$$

where $j\geqslant 1, 0\leqslant i\leqslant j$ are any fixed integers. Furthermore, we also consider the similar problem concerning the linear combinations of $\lambda _{f}(p^{j})$ and $\lambda _{g}(p^{j})$ in a given interval supported on the binary quadratic form $Q(\textbf{x})$. Similar problems for triple product L-functions are also studied.