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

TL;DR

This paper investigates the distribution of zeros of L-functions associated with cusp forms, providing new estimates for their density in certain regions, using classical analytic techniques.

Contribution

It offers a novel proof for zero-density estimates of L-functions attached to cusp forms, extending previous results with alternative methods.

Findings

Established new bounds for zero density in the critical strip.

Provided an alternative proof approach using classical methods.

Enhanced understanding of zero distribution for L-functions of cusp forms.

Abstract

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. 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 $\sigma \leq \Re s \leq 1$ for fixed $\sigma>1/2$ with respect to the imaginary part. We study estimates of \[ N_f(\sigma,T) = #\{\rho\in\mathbb{C} \mid L_f(\rho)=0, \sigma\ leq \Re\rho \leq 1, 0 \leq \Im\rho \leq T} \] for $1/2 \leq \sigma \leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivi\'{c}'s method.