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arXiv · 1310.0765

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

Yoshikatsu Yashiro
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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 $σ\leq \Re s \leq 1$ for fixed $σ>1/2$ 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 $1/2 \leq σ\leq1$ and large $T>0$. Using the methods of Karatsuba and Voronin we shall give another proof for Ivić's method.

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Yoshikatsu Yashiro. 2014-02-15. Zero-density estimates for L-functions attached to cusp forms. https://arxiv.org/abs/1310.0765

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