Zeros of L-functions outside the critical strip

Andrew R Booker; Frank Thorne

doi:10.2140/ant.2014.8.2027

Zeros of L-functions outside the critical strip

Andrew R Booker, Frank Thorne

Research output: Contribution to journal › Article (Academic Journal) › peer-review

6 Citations (Scopus)

226 Downloads (Pure)

Abstract

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if ƒ∈S_k(Γ₁(N)) is a classical holomorphic modular form whose L-function does not vanish for R(s)>(k+1)∕2, then ƒ is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree-1 L-functions.

Original language	English
Pages (from-to)	2027-2042
Number of pages	16
Journal	Algebra and Number Theory
Volume	8
Issue number	9
Early online date	28 Dec 2014
DOIs	https://doi.org/10.2140/ant.2014.8.2027
Publication status	Published - Dec 2014

Keywords

L-functions
Euler products
automorphic forms

Access to Document

10.2140/ant.2014.8.2027

Full-text PDF (accepted author manuscript)
This is the author accepted manuscript (AAM). The final published version (version of record) is available online via MSP at http://msp.org/ant/2014/8-9/p01.xhtml. Please refer to any applicable terms of use of the publisher.
Accepted author manuscript, 229 KB

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3 Finished

Detecting squarefree numbers
Booker, A. R.
1/07/13 → 1/07/15
Project: Research
L-functions and modular forms
Keating, J. P. & Booker, A. R.
1/06/13 → 30/09/19
Project: Research
Explicit number theory, automorphic forms and L-functions
Booker, A. R.
1/10/09 → 1/04/15
Project: Research

Professor Andrew R Booker
- Andrew.Booker@bristol.ac.uk
- School of Mathematics - Professor of Pure Mathematics
Person: Academic

Cite this

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abstract = "For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if ƒ∈Sk(Γ1(N)) is a classical holomorphic modular form whose L-function does not vanish for R(s)>(k+1)∕2, then ƒ is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree-1 L-functions. ",

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AB - For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if ƒ∈Sk(Γ1(N)) is a classical holomorphic modular form whose L-function does not vanish for R(s)>(k+1)∕2, then ƒ is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree-1 L-functions.

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KW - Euler products

KW - automorphic forms

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M3 - Article (Academic Journal)

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JO - Algebra and Number Theory

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Zeros of L-functions outside the critical strip

Abstract

Keywords

Access to Document

Detecting squarefree numbers

L-functions and modular forms

Explicit number theory, automorphic forms and L-functions

Professor Andrew R Booker

Cite this