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There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which provide another path for insights into number theory. Farkas and Kra, well-known masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for research-level problems. The text is suitable for a graduate course or for independent reading

## About book:

## About file:

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- Series:
**Graduate Studies in Mathematics 037** - Author:
**Hershel M. Farkas and Irwin Kra** - Year:
**2001** - Publisher:
**American Mathematical Society** - Language:
**English** - ISBN:
**0-8218-1392-7,9780821813928**

- DPI:
**600** - File size:
**5 266 741** - Format:
**djvu**

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There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Numb...

This book presents a cross-section of different aspects of Riemann surfaces, introducing the reader to the basics as well as highlighting new developments in the field. It provides a mixture of classical material, recent results and some non-mainstream to...

Most of the papers in this book deal with the theory of Riemann surfaces (moduli problems, automorphisms, etc.), abelian varieties, theta functions, and modular forms. Some of the papers contain surveys on the recent results in the topics of current inter...

It is gratifying to learn that there is new life in an old field that has been at the center of one's existence for over a quarter of a century. It is particularly pleasing that the subject of Riemann surfaces has attracted the attention of a new generati...

This book grew out of lectures on Riemann surfaces given by Otto Forster at the Universities of Munich, Regensburg and Munster. It provides an introduction to the subject, as well as presenting methods used in the study of complex manifolds in the special...

This text covers Riemann surface theory from elementary aspects to the fontiers of current research. Open and closed surfaces are treated with emphasis on the compact case, while basic tools are developed to describe the analytic, geometric, and algebraic...

This classic monograph is a self-contained introduction to the geometry of Riemann surfaces of constant curvature â€“1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater tha...

Few books on the subject of Riemann surfaces cover the relatively modern theory of dessins d'enfants (children's drawings), which was launched by Grothendieck in the 1980s and is now an active field of research. In this book, the authors begin with an ele...

As part of a series from the U. of Montreal promoting research in pure and applied mathematics, this volume constructs Riemann surfaces of infinite genus geometrically by pasting together plane domains and handles. In order to find a meaningful generaliza...

The United Nations Security Council has increasingly resorted to sanctions as part of its efforts to prevent and resolve conflict. Farrall traces the evolution of the Security Council's sanctions powers and charts the contours of the UN sanctions system. ...

Created for scholars, librarians, students and the general public, this work identifies and describes numerous reference books on American history. Arranged topically, subjects include politics and government, military, social and cultural items. Various ...

Unity is a powerful game development engine that provides rich functionalities to create 2D and 3D games. Unity 2D Game Development Cookbook is a practical guide to creating games with Unity. The book aims to serve the purpose of exploring problematic co...

Unity 3D is a powerful and popular game development tool that has taken game developers by storm. It is a cross-platform game engine, enabling you to write your game once and then port it to PCs, consoles, and even the Web, making it a great choice for bo...

What you will learn from this bookBuilding finite state machines (FSMs)Implementing a sensory systemApplying flocking behavior for a crowdExecuting your own A* pathfinding algorithm in Unity3DApplying random and probability techniques in a betting gameUsi...

Make your characters interact with buttons and program triggered action sequences. Create custom characters and code dynamic objects and players' interaction with them. Synchronize movement of character and environmental objects. Add and control animation...