Zero-density estimates for L-functions attached to cusp forms

Abstract

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $S{L}_2(\mathbb{Z})$. Let $f\in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the distribution of zeros of $L_f(s)$ in the strip $\sigma \le \Re s\le 1$ for fixed $\sigma >1/2$ with respect to the imaginary part. We study estimates of \[ N_f(σ,T) = #\{ρ\in\mathbb{C} \mid L_f(ρ)=0, σ leq \Reρ\leq 1, 0 \leq \Imρ\leq T} \] for $1/2\le \sigma \le 1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivić's method.

arXiv comment21 pages