Spectral Methods of Automorphic Forms
Spectral Methods of Automorphic Forms

Author: Henryk Iwaniec

Publisher: American Mathematical Society, Revista Matemática Iberoamericana (RMI), Madrid, Spain

Published: 2021-11-17

Total Pages: 220

ISBN-13: 1470466228

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Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style. The first edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references. Iwaniec treats the spectral theory of automorphic forms as the study of the space of $L^2$ functions on the upper half plane modulo a discrete subgroup. Key topics include Eisenstein series, estimates of Fourier coefficients, Kloosterman sums, the Selberg trace formula and the theory of small eigenvalues. Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contributions to number theory.

Ramanujan’s Notebooks
Ramanujan’s Notebooks

Author: Bruce C. Berndt

Publisher: Springer Science & Business Media

Published: 1999-01-18

Total Pages: 384

ISBN-13: 9780387967943

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During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

Topics in Classical Automorphic Forms
Topics in Classical Automorphic Forms

Author: Henryk Iwaniec

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 274

ISBN-13: 0821807773

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This volume discusses various perspectives of the theory of automorphic forms drawn from the author's notes from a Rutgers University graduate course. In addition to detailed and often nonstandard treatment of familiar theoretical topics, the author also gives special attention to such subjects as theta- functions and representatives by quadratic forms. Annotation copyrighted by Book News, Inc., Portland, OR

Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory
Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory

Author: Solomon Friedberg

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 320

ISBN-13: 0821839632

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Multiple Dirichlet series are Dirichlet series in several complex variables. A multiple Dirichlet series is said to be perfect if it satisfies a finite group of functional equations and has meromorphic continuation everywhere. The earliest examples came from Mellin transforms of metaplectic Eisenstein series and have been intensively studied over the last twenty years. More recently, many other examples have been discovered and it appears that all the classical theorems on moments of $L$-functions as well as the conjectures (such as those predicted by random matrix theory) can now be obtained via the theory of multiple Dirichlet series. Furthermore, new results, not obtainable by other methods, are just coming to light. This volume offers an account of some of the major research to date and the opportunities for the future. It includes an exposition of the main results in the theory of multiple Dirichlet series, and papers on moments of zeta- and $L$-functions, on new examples of multiple Dirichlet

Frontiers in Number Theory, Physics, and Geometry II
Frontiers in Number Theory, Physics, and Geometry II

Author: Pierre E. Cartier

Publisher: Springer Science & Business Media

Published: 2007-07-18

Total Pages: 806

ISBN-13: 3540303081

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Ten years after a 1989 meeting of number theorists and physicists at the Centre de Physique des Houches, a second event focused on the broader interface of number theory, geometry, and physics. This book is the first of two volumes resulting from that meeting. Broken into three parts, it covers Conformal Field Theories, Discrete Groups, and Renormalization, offering extended versions of the lecture courses and shorter texts on special topics.

Modular Forms
Modular Forms

Author: Henri Cohen

Publisher: American Mathematical Soc.

Published: 2017-08-02

Total Pages: 714

ISBN-13: 0821849476

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The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and “fun” subject in itself and abounds with an amazing number of surprising identities. This comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. After a number of motivating examples such as elliptic functions and theta functions, the modular group, its subgroups, and general aspects of holomorphic and nonholomorphic modular forms are explained, with an emphasis on explicit examples. The heart of the book is the classical theory developed by Hecke and continued up to the Atkin–Lehner–Li theory of newforms and including the theory of Eisenstein series, Rankin–Selberg theory, and a more general theory of theta series including the Weil representation. The final chapter explores in some detail more general types of modular forms such as half-integral weight, Hilbert, Jacobi, Maass, and Siegel modular forms. Some “gems” of the book are an immediately implementable trace formula for Hecke operators, generalizations of Haberland's formulas for the computation of Petersson inner products, W. Li's little-known theorem on the diagonalization of the full space of modular forms, and explicit algorithms due to the second author for computing Maass forms. This book is essentially self-contained, the necessary tools such as gamma and Bessel functions, Bernoulli numbers, and so on being given in a separate chapter.

Partitions, q-Series, and Modular Forms
Partitions, q-Series, and Modular Forms

Author: Krishnaswami Alladi

Publisher: Springer Science & Business Media

Published: 2011-11-01

Total Pages: 233

ISBN-13: 1461400287

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Partitions, q-Series, and Modular Forms contains a collection of research and survey papers that grew out of a Conference on Partitions, q-Series and Modular Forms at the University of Florida, Gainesville in March 2008. It will be of interest to researchers and graduate students that would like to learn of recent developments in the theory of q-series and modular and how it relates to number theory, combinatorics and special functions.