Canonical products generated by perturbations of the sequence of integers, and their asymptotic estimation

We obtain asymptotic estimates for canonical products with complex zeros of the form $\lambda_n=n+o(n)$. A formula is found for the excess of the system of exponentials $\{e^{i\lambda_nt}\}_{n\in\mathbb{Z}}$ in the space $L^2(-\pi,\pi)$. We consider some particular cases of sequences $\{\lambda_n\}_{n\in\mathbb{Z}}$.