A new zero-density result of \(L\)-functions attached to Maass forms
type: Item
instance of: scholarly article
title: A new zero-density result of \(L\)-functions attached to Maass forms
zbMATH Open document ID: 1266.11091
author: Zhao Xu
publication date: 2011-06-24T00:00:00Z
review text: If \(f\) is a Maass cusp form for \(\text{SL}(2, \mathbb Z)\) which is an eigenform for the Hecke operators \(T(n)\) and the reflection operator \(T: z \mapsto -\overline{z},\) then the associated \(L\)-function is defined as \[ L(f,s) = \sum_{n=1}^{\infty} \frac{\lambda(n)}{n^s} \] where \(T(n)f = \lambda(n) f\) and \(\lambda (1) = 1.\) Denote by \(N_L(\sigma, T)\) the number of zeros of \(L(f,s)\) with real part greater than or equal to \(\sigma\) and imaginary part having absolute value not exceeding \(T.\) The author shows that the result of \textit{A. Sankaranarayanan} and \textit{J. Sengupta} from [Acta Arith. 127, No. 3, 273--284 (2007; Zbl 1137.11035)] to the effect that for \(\sigma \geq \frac{1}{2}+\frac{1}{\log T}\), \[ N_L(\sigma, T) \ll T^{\frac{4(1-\sigma)}{3-2\sigma}}\log ^{26} T \] can be improved for \(\frac{3}{4}\leq\sigma \leq 1\) to \[ N_L(\sigma, T) \ll T^{\frac{(1-\sigma)(8 \sigma - 5)}{-2 \sigma^2 + 6 \sigma -3}}\log ^{57} T. \] As in the result of Sankaranarayanan and Sengupta, the implied constant depends on \(f\).
reviewed by: Sheldon T. Joyner
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11N75
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 5912925
zbMATH Keywords: zero-density
zbMATH Keywords: automorphic forms
zbMATH Keywords: \(L\)-functions
MaRDI profile type: Publication
full work available at URL: s10114-011-8310-0
OpenAlex ID: W2116867646
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DOI: 10.1007/S10114-011-8310-0
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 5912925