Non-vanishing of Rankin-Selberg convolutions for Hilbert modular forms

In this paper, we study the non-vanishing of the central values of the Rankin-Selberg L-function of two adelic Hilbert primitive forms $$\mathbf{f}$$ and $$\mathbf{g}$$ , both of which have varying weight parameter k. We prove that, for sufficiently large $$k\in 2{\mathbb {N}}^n$$ , there are at least $$\frac{\mathrm{N}(k)}{\log ^c \mathrm{N}(k)}$$ adelic Hilbert primitive forms $$\mathbf{f}$$ of weight k for which $$L(\frac{1}{2}, \mathbf{f}\otimes \mathbf{g})$$ are nonzero.