Moments Of Products Of L Functions Dissertation

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Moments of Products of L-functions Dissertation

Author : Caroline LaRoche Turnage-Butterbaugh
Publisher :
ISBN 13 :
Total Pages : 100 pages
Book Rating : 4.:/5 (892 download)

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Book Synopsis Moments of Products of L-functions Dissertation by : Caroline LaRoche Turnage-Butterbaugh

Download or read book Moments of Products of L-functions Dissertation written by Caroline LaRoche Turnage-Butterbaugh and published by . This book was released on 2014 with total page 100 pages. Available in PDF, EPUB and Kindle. Book excerpt: We first consider questions on the distribution of the primes. Using the recent advancement towards the Prime k -tuple Conjecture by Maynard and Tao, we show how to produce infinitely many strings of consecutive primes satisfying specified congruence conditions. We answer an old question of Erdos and Turan by producing strings of consecutive primes whose successive gaps form an increasing (respectively decreasing) sequence. We also show that such strings exist whose successive gaps follow a certain divisibility pattern. Finally, for any coprime integers a and D ≥ 1, we refine a theorem of D. Shiu and find strings of consecutive primes of arbitrary length in the congruence class a mod D. These results were proved jointly with William D. Banks and Tristan Freiberg. We next consider the vertical distribution of the nontrivial zeros of certain Dedekind zeta-functions. In particular, let K be a quadratic number field. Using the mixed second moments of derivatives of the Dedekind zeta-function attached to K on the critical line, we prove the existence of gaps between consecutive zeros of the Dedekind zeta-function attached to K on the critical line which are at least 2.44949... times the average spacing. Finally, assuming the Generalized Riemann Hypothesis and some standard conjectures, we prove upper bounds for moments of arbitrary products of automorphic L -functions and for Dedekind zeta-functions of Galois number fields on the critical line. As an application, we use these bounds to estimate the variance of the coefficients of these zeta- and L -functions in short intervals. We also prove upper bounds for moments of products of central values of automorphic L -functions twisted by quadratic Dirichlet characters and averaged over fundamental discriminants. These results were proved jointly with Micah B. Milinovich.

Moments of Automorphic L-Functions

Author : Ming-Ho Ng
Publisher :
ISBN 13 : 9781361031278
Total Pages : pages
Book Rating : 4.0/5 (312 download)

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Book Synopsis Moments of Automorphic L-Functions by : Ming-Ho Ng

Download or read book Moments of Automorphic L-Functions written by Ming-Ho Ng and published by . This book was released on 2017-01-26 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation, "Moments of Automorphic L-functions" by Ming-ho, Ng, 吳銘豪, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: This thesis is devoted to investigation of moments of automorphic L-functions, especially on the central values or the edges of the critical strip of automorphic L-functions. There are nine chapters. Chapter 1 is an introduction and provides some background on the analytic theory of automorphic forms. Chapters 2, 3, 4, 5 and 6 are about L-functions associated to the holomorphic cusp forms, while Chapters 7, 8 and 9 are focused on the L-functions associated to the Maass forms. Chapter 2 is the study of the first moment of the symmetric-square L-functions associated to the holomorphic cusp forms. Asymptotic formulae for the twisted first moment of central values of the symmetric-square L-functions with harmonic weight, in the weight aspect are obtained. The result (in Theorem 2.1.1) extends and improves the known results in the literature. As an application, it is applied to derive an asymptotic formula for the first moment of central values of the symmetric-square L-functions without harmonic weight, under the assumption of the non-negativity of symmetric-square L-functions at the center of critical strip. Analogous new formulae without harmonic weight of the first and second moments of the Hecke L-functions are proved in Chapter 3. Unlike the case in Chapter 2, the results in Chapter 3 are unconditional. In Chapter 4, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the symmetric square L-functions, with or without harmonic weight, are investigated in the weight aspect. The similar problem in the level aspect was treated by Lau, Royer and Wu. The theme of Chapter 5, as well as Chapter 4, is to examine, in the weight aspect, complex moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the central values of the square L-functions or the square of L-functions, with or without harmonic weight. All the above mentioned results, appeared in Chapters 4 and 5, are in the asymptotic form, that is, given by a formula consisting of a main term and an error term. Chapter 6 investigates the asymptotic behavior of the main terms of the results in Chapters 4 and 5. In particular, precise expansions of high moments are given. In Chapters 7, 8 and 9, the previous studies are carried over to Maass forms in the spectral aspect. The first two moments of central values of symmetric square L-functions associated to Maass forms are computed in Chapter 7. The first four moments of central values of L-functions associated to Maass forms are obtained in Chapter 8. Chapter 9 is to research the mixed moments of central values of symmetric square L-functions twisted by the central values of L-functions or the square of L-functions. These investigations for Maass form are not yet done in the literature. Subjects: L-functions Automorphic functions

The Behaviour of L-functions at the Edge of the Critical Strip and Applications

Author : Xiannan Li
Publisher : Stanford University
ISBN 13 :
Total Pages : 99 pages
Book Rating : 4.F/5 ( download)

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Book Synopsis The Behaviour of L-functions at the Edge of the Critical Strip and Applications by : Xiannan Li

Download or read book The Behaviour of L-functions at the Edge of the Critical Strip and Applications written by Xiannan Li and published by Stanford University. This book was released on 2011 with total page 99 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large number of problems in number theory can be reduced to statements about L-functions. In this thesis, we study L-functions at the edge of the critical strip, and relate these to a variety of objects of arithmetic interest.

Upper Bounds and Moments of L-functions

Author : Vorrapan Chandee
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (664 download)

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Book Synopsis Upper Bounds and Moments of L-functions by : Vorrapan Chandee

Download or read book Upper Bounds and Moments of L-functions written by Vorrapan Chandee and published by . This book was released on 2010 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: L-functions are some of the most studied objects in number theory. Although many crucial properties of L-functions remain mysterious, central conjectures such as the generalized Riemann hypothesis (GRH). This thesis concerns properties of L-functions. In particular, we focus on studying upper bounds and moments of $L$-functions. Assuming GRH, we give effective explicit upper bounds for L-functions on the critical line and apply these bounds to determine what numbers are represented by a given ternary quadratic form. Moreover the best known version of the Lindelof hypothesis from the Riemann hypothesis (RH) is also derived. Another important way of understanding LH is through moments of L-functions. Information about moments sheds light on the distribution of values of \zeta(1/2 + it). We try to understand the joint distribution of quantities like \zeta(1/2 + it) and \zeta(1/2 + it + i). To study these we consider "shifted moments" of the zeta function and obtain good upper and lower estimates for such moments.

On the Moments of Central Values of Modular L-Functions

Author : Benjamin Justus
Publisher : LAP Lambert Academic Publishing
ISBN 13 : 9783838338224
Total Pages : 80 pages
Book Rating : 4.3/5 (382 download)

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Book Synopsis On the Moments of Central Values of Modular L-Functions by : Benjamin Justus

Download or read book On the Moments of Central Values of Modular L-Functions written by Benjamin Justus and published by LAP Lambert Academic Publishing. This book was released on 2010-12-01 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt: The thesis studies the integer-power moments of the central values of families of modular L-functions. The two families under consideration in the thesis are those quadratic twists of a L-function associated with a cusp from and L-functions of a Hecke-basis of the space of cusp forms. Applications of the moment estimates derived in the thesis include (1) a non-vanishing result (2) a zero density estimate for quadratic twisted L-functions.

Moments of Automorphic L-functions and Related Problems

Author : Ian Petrow
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (853 download)

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Book Synopsis Moments of Automorphic L-functions and Related Problems by : Ian Petrow

Download or read book Moments of Automorphic L-functions and Related Problems written by Ian Petrow and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different moments of central values of L-functions of a fixed GL_2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f \otimes \chi_d) is -1, the central value vanishes and one instead studies the derivative L'(1/2, f \otimes \chi_d). We prove two theorems in the root number -1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL_2 objects. We study the family of cusp forms of level q^2 which are given by f \otimes \chi, where f is a modular form of prime level q and \chi is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted convolution sums of Fourier coefficients of full-level GL_2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average over shifts is proportional to the length of the shifted sum. The asymptotic in this range depends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions.

Value-Distribution of L-Functions

Author : Jr̲n Steuding
Publisher : Springer Science & Business Media
ISBN 13 : 3540265260
Total Pages : 320 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Value-Distribution of L-Functions by : Jr̲n Steuding

Download or read book Value-Distribution of L-Functions written by Jr̲n Steuding and published by Springer Science & Business Media. This book was released on 2007-06-06 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes present recent results in the value-distribution theory of L-functions with emphasis on the phenomenon of universality. Universality has a strong impact on the zero-distribution: Riemann’s hypothesis is true only if the Riemann zeta-function can approximate itself uniformly. The text proves universality for polynomial Euler products. The authors’ approach follows mainly Bagchi's probabilistic method. Discussion touches on related topics: almost periodicity, density estimates, Nevanlinna theory, and functional independence.

The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions

Author : Valentin Blomer
Publisher : American Mathematical Society
ISBN 13 : 1470456788
Total Pages : 160 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions by : Valentin Blomer

Download or read book The Second Moment Theory of Families of $L$-Functions–The Case of Twisted Hecke $L$-Functions written by Valentin Blomer and published by American Mathematical Society. This book was released on 2023-02-13 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Moments and Zeros of L-functions

Author : Quanli Shen
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (13 download)

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Book Synopsis Moments and Zeros of L-functions by : Quanli Shen

Download or read book Moments and Zeros of L-functions written by Quanli Shen and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We study moments and zeros of L-functions in this thesis. In Chapter 2, by following closely Soundararajan-Young's method, we prove an asymptotic for the fourth moment of quadratic Dirichlet L-functions under the generalized Riemann hypothesis. Unconditionally, we are able to give a sharp lower bound that agrees with Keating-Snaith's conjecture. In Chapter 3, we use a recursive method that was pioneered by Heath-Brown and developed by Young to give an asymptotic with an error O(X1/2+E) for the smoothed first moment of quadratic twists of modular L-functions. The result is analogous to Sono's work on the second moment of quadratic Dirichlet L-functions. It improves previous results of Iwaniec and Soundararajan-Radziwill. In Chapter 4, we obtain an explicit result for the number of zeros, in a box, of Dedekind zeta functions, which improves a result of Trudgian. Our argument is based on previous works of Bennett-Martin-O'Bryant-Rechnitzer, Kadiri-Ng and Trudgian.

Discrete Moments and Linear Combinations of L-functions

Author : Scott J. Kirila
Publisher :
ISBN 13 :
Total Pages : 74 pages
Book Rating : 4.:/5 (15 download)

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Book Synopsis Discrete Moments and Linear Combinations of L-functions by : Scott J. Kirila

Download or read book Discrete Moments and Linear Combinations of L-functions written by Scott J. Kirila and published by . This book was released on 2018 with total page 74 pages. Available in PDF, EPUB and Kindle. Book excerpt: "In the first half of this thesis, assuming the Riemann hypothesis, we establish an upper bound for the 2k-th discrete moment of the derivative of the Riemann zeta-function at nontrivial zeros, where k is a positive real number. Our upper bound agrees with conjectures of Gonek and Hejhal and of Hughes, Keating, and O'Connell. This sharpens a result of Milinovich. Our proof builds upon a method of Adam Harper concerning continuous moments of the zeta-function on the critical line. We also prove similar estimates for higher derivatives of the zeta-function. In the second half, we consider how often two distinct linear combinations of L-functions can have a common zero in one of three regions: the critical line Re(s) = 1/2 , vertical strips contained within the right-half of the critical strip, and to the right of the line Re(s) = 1. On the critical strip, we show that, under certain hypotheses, at least 1/3 of the nontrivial zeros of the Riemann zeta-function are not zeros of a linear combination of two Dirichlet L-functions. In the remaining two regions, we prove that a positive proportion of the zeros of a linear combination [formula would not render] are not zeros of [formula would not render] provided some reasonable conditions on the characters [formula would not render] and coefficients an; bm are met."--Page vii.

On the Moments of Central Values of Modular L-functions

Author : Benjamin Justus
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (462 download)

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Book Synopsis On the Moments of Central Values of Modular L-functions by : Benjamin Justus

Download or read book On the Moments of Central Values of Modular L-functions written by Benjamin Justus and published by . This book was released on 2008 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The thesis studies the integer-power moments of the central values of families of modular L -functions. The two families under consideration in the thesis are those quadratic twists of a L -function associated with a cusp form and L -functions of a Hecke-basis of the space of cusp forms. Appropriate moment estimates are derived for each family. Applications of the derived estimates are given.

Automorphic Forms on GL (3,TR)

Author : D. Bump
Publisher : Springer
ISBN 13 : 3540390553
Total Pages : 196 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Automorphic Forms on GL (3,TR) by : D. Bump

Download or read book Automorphic Forms on GL (3,TR) written by D. Bump and published by Springer. This book was released on 2006-12-08 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Fourth Moment of Automorphic L-functions of Prime Power Level

Author : Olga Balkanova
Publisher :
ISBN 13 :
Total Pages : 128 pages
Book Rating : 4.:/5 (111 download)

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Book Synopsis The Fourth Moment of Automorphic L-functions of Prime Power Level by : Olga Balkanova

Download or read book The Fourth Moment of Automorphic L-functions of Prime Power Level written by Olga Balkanova and published by . This book was released on 2015 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main result of this dissertation is an asymptotic formula for the fourth moment of automorphic L-functions of prime power level [rho vu], [vu] --> [infinity]. This is a continuation of the work of Rouymi, who computed the first three moments at prime power level, and a generalisation of results obtained for prime level by Duke, Friedlander & Iwaniec and Kowalski, Michel & Vanderkam.

Families and Statistics of L-functions

Author : Joshua Stucky
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (139 download)

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Book Synopsis Families and Statistics of L-functions by : Joshua Stucky

Download or read book Families and Statistics of L-functions written by Joshua Stucky and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first chapter of this dissertation provides a general introduction to the study of families of L-functions along with the necessary tools for understanding their behavior. In particular, we introduce the families studied in the second and third chapters of this dissertation and provide some prerequisite knowledge on these families. The second chapter of this dissertation studies a family of L-functions attached to Hecke Grossencharacters and extends a geometric result of Ricci concerning the equidistribution of prime ideals of Z[i] in narrow sectors. The third chapter of this dissertation studies a family of L-functions attached to automorphic forms on GL2. Specifically, we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL2 with respect to a congruence subgroup [gamma]1(q). We improve on previous work and obtain an unconditional upper bound of the correct order of magnitude.

Moments of Cubic Hecke L-Functions

Author : Arihant Jain
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (139 download)

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Book Synopsis Moments of Cubic Hecke L-Functions by : Arihant Jain

Download or read book Moments of Cubic Hecke L-Functions written by Arihant Jain and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Moments of families of L-functions provide understanding of their size and also about their distribution. The aim of this thesis is to calculate the asymptotics of the first moment of L-functions associated to primitive cubic Hecke characters over $Q(\omega)$ and upper bounds for 2k-th moments for the same family. Both of these results assume Generalized Riemann Hypothesis. We consider the full family of characters which results in a main term of order x log x. We also calculate conditional upper bounds for 2k-th moments for the same family and conclude that there ” x primitive characters of conductor at most x for which the L-function doesn't vanish at the central point.

Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

Author : Daniel Alpay
Publisher : Birkhäuser
ISBN 13 : 3319688499
Total Pages : 501 pages
Book Rating : 4.3/5 (196 download)

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Book Synopsis Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations by : Daniel Alpay

Download or read book Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations written by Daniel Alpay and published by Birkhäuser. This book was released on 2018-01-30 with total page 501 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, which is dedicated to Heinz Langer, includes biographical material and carefully selected papers. Heinz Langer has made fundamental contributions to operator theory. In particular, he has studied the domains of operator pencils and nonlinear eigenvalue problems, the theory of indefinite inner product spaces, operator theory in Pontryagin and Krein spaces, and applications to mathematical physics. His works include studies on and applications of Schur analysis in the indefinite setting, where the factorization theorems put forward by Krein and Langer for generalized Schur functions, and by Dijksma-Langer-Luger-Shondin, play a key role. The contributions in this volume reflect Heinz Langer’s chief research interests and will appeal to a broad readership whose work involves operator theory.

Moments of Automorphic L-functions at Special Points

Author : Alexander Lu Beckwith
Publisher :
ISBN 13 :
Total Pages : 173 pages
Book Rating : 4.:/5 (123 download)

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Book Synopsis Moments of Automorphic L-functions at Special Points by : Alexander Lu Beckwith

Download or read book Moments of Automorphic L-functions at Special Points written by Alexander Lu Beckwith and published by . This book was released on 2020 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the behavior of families of L-functions at exhibiting conductor-dropping behavior. We will derive asymptotic expansions of the short interval first and second moments of GL(2)xGL(2) L-functions at special points with power-saving error terms. As a consequence, we show that large number of cusp forms for Hecke congruence surfaces of prime level are simultaneously destroyed in two directions of the associated Teichmuller space. We also establish upper bounds for the second moment of GL(2)xGL(3) L-functions and the sixth moment of GL(2) L-functions at special points as the spectral parameter varies in a short interval.