On Li's coefficients for the Rankin-Selberg \(L\)-functions
type: Item
instance of: scholarly article
title: On Li's coefficients for the Rankin-Selberg \(L\)-functions
zbMATH Open document ID: 1248.11036
author: Almasa Odžak
author: Lejla Smajlovic
published in: The Ramanujan Journal
publication date: 2010-05-19T00:00:00Z
review text: In 1997, \textit{X.-J. Li} [J. Number Theory 65, No. 2, 325--333 (1997; Zbl 0884.11036)] studied the sequence \[ \lambda_n = \lim_{T\to \infty} \sum_{|\text{Im}\,\rho |\leq T}\left(1-\left(1-\frac1\rho\right)^n\right), \] where \(\rho\) runs over non-trivial zeros of the Riemann zeta-function. (The first step is to show that \(\lambda_n\) is well-defined, i.e., that the limit indeed converges, for all \(n\).) Li showed that the non-negativity of \(\lambda_n\) for all \(n\) is equivalent to the Riemann hypothesis. The numbers \(\lambda_n, \;n=1, 2, \dots\) have since become known as Li coefficients. By replacing the Riemann zeta-function with some other automorphic \(L\) function, one may define generalized Li coefficients. The present paper is concerned with generalized Li coefficients attached to the Rankin-Selberg convolution \(L\) function \(L(s, \pi \times \pi'),\) where \(\pi\) and \(\pi'\) are irreducible (unitary) cuspidal automorphic representations of \(\text{GL}_m(\mathbb A_F)\) and \(\text{GL}_{m'}(\mathbb A_F)\) and \(\mathbb A_F\) is the adele ring of a number field \(F.\) The authors first give an estimate for the number of nontrivial zeros of \(L(s, \pi \times \pi')\) up to height \(T\) and a version of the Weil explicit formula for \(L(s, \pi \times \pi').\) This is then used to prove the well-definedness of the the generalized Li coefficients (i.e., convergence of the above limit in this case). The next step is to give an arithmetic expression for the generalized Li coefficients, extending the work of \textit{E. Bombieri} and \textit{J. C. Lagarias} [J. Number Theory 77, No. 2, 274--287 (1999; Zbl 0972.11079)] for the original Li coefficients, and \textit{J. Lagarias} [Ann. Inst. Fourier 57, No. 5, 1689--1740 (2007; Zbl 1216.11078)] for generalized Li coefficients attached to \(\text{GL}_n\) standard \(L\) functions. This is done in two different ways, and includes a decomposition of the generalized Li coefficients into finite and archimedean contri…
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5709208
zbMATH Keywords: Li's coefficients
zbMATH Keywords: Rankin-Selberg \(L\)-functions
zbMATH Keywords: generalized Riemann hypothesis
zbMATH Keywords: generalized Ramanujan conjecture
reviewed by: Joseph Hundley
MaRDI profile type: Publication
full work available at URL: s11139-009-9175-z
OpenAlex ID: W1991158871
cites work: Q5558293
cites work: Q4469294
cites work: Explicit formula for a fundamental class of functions
cites work: A Note on Weil’s Explicit Formula
cites work: On A. Weil's explicit formula.
cites work: Complements to Li's criterion for the Riemann Hypothesis
cites work: Li's criterion and zero-free regions of \(L\)-functions
cites work: Q5439483
cites work: Boundedness of automorphic 𝐿–functions in vertical strips
cites work: Q4830109
cites work: On Euler Products and the Classification of Automorphic Representations I
cites work: On Euler Products and the Classification of Automorphic Forms II
cites work: Q4321409
cites work: Li coefficients for automorphic \(L\)-functions
cites work: The positivity of a sequence of numbers and the Riemann hypothesis
cites work: Explicit formulas for Dirichlet and Hecke \(L\)-functions
cites work: An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials
cites work: Q4263342
cites work: Le spectre résiduel de ${\rm GL}(n)$
cites work: Zeros of principal \(L\)-functions and random matrix theory
cites work: On Certain L-Functions
cites work: Fourier Transforms of Intertwining Operators and Plancherel Measures for GL(n)
cites work: Local coefficients as Artin factors for real groups
cites work: A proof of Langlands' conjecture on Plancherel measures; complementary series for \(p\)-adic groups
cites work: On Li's criterion for the Riemann hypothesis for the Selberg class
cites work: Sharpenings of Li's criterion for the Riemann hypothesis
DOI: 10.1007/S11139-009-9175-Z
Recommended article: On τ-Li Coefficients for Rankin–Selberg L-Functions
Recommended article: On the \(\tau \)-Li coefficients for automorphic \(L\)-functions
Recommended article: On asymptotic behavior of generalized Li coefficients in the Selberg class
Recommended article: On relations equivalent to the generalized Riemann hypothesis for the Selberg class
Recommended article: Li coefficients for automorphic \(L\)-functions
Recommended article: Q3089321
Recommended article: The generalized prime number theorem for automorphic \(L\)-functions
Recommended article: The saddle-point method and the Li coefficients
Recommended article: On asymptotic behavior of generalized Li coefficients
Recommended article: On the asymptotic criterion for the zero-free regions of certain $L$-functions
description: scientific article; zbMATH DE number 5709208