Character sums, automorphic forms, equidistribution, and Ramanujan graphs Part I. The Kloosterman sum conjecture over function fields

2003 ◽  
Vol 15 (5) ◽  
Author(s):  
Ching-Li Chai ◽  
Wen-Ching Winnie Li
Keyword(s):  
Automorphic Forms ◽  
Function Fields ◽  
Character Sums ◽  
Kloosterman Sum ◽  
Ramanujan Graphs

scholarly journals Ramanujan graphs and exponential sums over function fields

2020 ◽  
Vol 217 ◽  
pp. 44-77
Author(s):  
Naser T. Sardari ◽  
Masoud Zargar
Keyword(s):  
Exponential Sums ◽  
Function Fields ◽  
Ramanujan Graphs

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

scholarly journals Mean values of derivatives of L-functions in function fields: III

2018 ◽  
Vol 149 (04) ◽  
pp. 905-913
Author(s):  
Julio Andrade
Keyword(s):  
Functional Equation ◽  
Function Fields ◽  
Character Sums ◽  
Asymptotic Formulas ◽  
Second Derivative ◽  
Approximate Functional Equation ◽  
Ground Field ◽  
Mean Values ◽  
First Moment ◽  
Derivatives Of

AbstractIn this series of papers, we explore moments of derivatives of L-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet L-functions over function fields when the average is taken over monic and irreducible polynomials P in 𝔽q[T]. When the cardinality q of the ground field is fixed and the degree of P gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of L-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.

scholarly journals Automorphic Forms and Sums of Squares over Function Fields

1999 ◽  
Vol 79 (2) ◽  
pp. 301-329 ◽  
Author(s):  
Jeffrey Hoffstein ◽  
Kathy D. Merrill ◽  
Lynne H. Walling
Keyword(s):  
Automorphic Forms ◽  
Function Fields ◽  
Sums Of Squares

scholarly journals Character sums and abelian Ramanujan graphs

1992 ◽  
Vol 41 (2) ◽  
pp. 199-217 ◽  
Author(s):  
Wen-Ch'ing Winnie Li ◽  
Keqin Feng
Keyword(s):  
Character Sums ◽  
Ramanujan Graphs

scholarly journals On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Xiaoling Xu
Keyword(s):  
Character Sums ◽  
Gauss Sums ◽  
Dedekind Sums ◽  
Character Sum ◽  
Kloosterman Sum ◽  
Power Mean ◽  
Computational Problem ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.

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