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International Mathematics Research Notices Advance Access published online on October 3, 2009
International Mathematics Research Notices, doi:10.1093/imrn/rnp148
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© The Author 2009. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

Upper Bounds on L-Functions at the Edge of the Critical Strip

Xiannan Li

Department of Mathematics, Stanford University, Stanford, CA 94305, USA

Correspondence: Correspondence to be sent to: xli@math.stanford.edu

Received for publication February 2, 2009. Revision received July 19, 2009. Accepted for publication August 21, 2009.

The first 150 words of the full text of this article appear below.


   1. Introduction
 
In this article, we are concerned with establishing bounds for L(1) where L(s) is a general L-function, and specifically, we shall be most interested in the case where no good bound for the size of the coefficients of the L-function is known. In this case, results are available due to Iwaniec [9, 10], and Molteni [16], but this type of investigation is still in its infancy, and the limitations of the methods available to analytic number theorists are unclear.

The value of L(1) has historically been one of great interest. The first interesting examples of bounds for L(1) come from Dirichlet L-functions with nonprincipal character. Let L(s, {chi}) denote a Dirichlet L-function where {chi} is a nonprincipal Dirichlet character with modulus q. In the case that {chi} is not a real quadratic character, . . . [Full Text of this Article]

1.1 Application to coefficients of Maass forms
Corollary 1
1.2 Lower and upper bounds for Maass forms
1.3 Convexity bounds for Rankin–Selberg L-functions
1.4 Overview of article
1.5 Formal definitions
Theorem 2

   2. Proof of Theorem 2
 
2.1 An bound for log L(s)
Lemma 1
Proof
Lemma 2
Proof
Lemma 3
Proof
2.2 Proof of the theorem
Lemma 4
Proof
Proof

   3. Rankin–Selberg and Refined Upper Bounds
 
Theorem 3
Corollary 4
Remark 1
Corollary 5
Corollary 6
3.1 Proof of Theorem 3
Proof
3.2 Upper bounds and symmetric powers
Proof
Proof
3.3 Proof of Corollaries 5 and 6
Proof
Proof

   4. Lower Bounds
 
Corollary 7
Lemma 5
Proof
Proof
Remark 2

   5. Some Remarks on GRH
 

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