Distribution of zeros and zero-density estimates for the derivatives of $L$-functions attached to cusp forms

TL;DR

This paper investigates the distribution of zeros and zero-density estimates for derivatives of $L$-functions associated with cusp forms, using advanced analytic techniques to relate zeros of the functions and their derivatives.

Contribution

It establishes a relation between zeros of $L_f(s)$ and its derivatives and provides new zero-density estimates for these derivatives.

Findings

Relation between zeros of $L_f(s)$ and its derivatives

Zero-density estimates for derivatives of $L_f(s)$

Application of Berndt's and Littlewood's methods

Abstract

Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros of $L_f(s)$ and the derivatives of $L_f(s)$ using Berndt's method, and an estimate of zero-density of the derivatives of $L_f(s)$ based on Littlewood's method.