Complex Analysis with Applications to Number Theory

Tarlok Nath Shorey

Description

Focuses on interactions of complex analysis with number theory

Supplements suitable solved examples and problems with all chapters

Is authored by the winner of the Shanti Swarup Bhatnagar Prize for Science and Technology

The book discusses major topics in complex analysis with applications to number theory. This book is intended as a text for graduate students of mathematics and undergraduate students of engineering, as well as to researchers in complex analysis and number theory. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two and three manifolds and number theory. In additional to solved examples and problems, the book covers most of the topics of current interest, such as Cauchy theorems, Picard’s theorems, Riemann–Zeta function, Dirichlet theorem, gamma function and harmonic functions. 

Introduction And Preliminaries.- Cauchy Theorems and Their Applications.-  Conformal Mappings and Riemann Mapping Theorem.- Picard's Theorems.- Factorisation of Analytic Functions in C and in a Region.- Gamma Function.- Riemann Zeta Function.- Dirichlet Series and Dirichlet Theorem.- Harmonic Functions.- Elliptic Functions and Modular Forms.

Sujets

• Sciences formelles
• Mathématiques générales
• Mathématiques générales
• Mathématiques fondamentales
• Mathématiques pures
• B
• Analysis
• Mathematics
• Number theory
• Sciences et techniques
• Mathematical Analysis

Informations

• Publié par: Springer Singapore
• Date de parution: 13 novembre 2020
• Nombre de lectures: 0
• EAN13: 9789811590979
• Langue: English
• Poids de l'ouvrage: 6 Mo

Extrait

Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences
Series Editors
Irene Fonseca

Carnegie Mellon University, Pittsburgh, PA, USA Gopal Prasad

University of Michigan, Ann Arbor, USA
Editorial Board
Manindra Agrawal

Indian Institute of Technology Kanpur, Kanpur, India Weinan E

Princeton University, Princeton, USA Chandrashekhar Khare

University of California, Los Angeles, USA Mahan Mj

Tata Institute of Fundamental Research, Mumbai, India Ritabrata Munshi

Tata Institute of Fundamental Research, Mumbai, India S. R. S Varadhan

New York University, New York, USA

The Infosys Science Foundation Series in Mathematical Sciences, a Scopus-indexed book series, is a sub-series of the Infosys Science Foundation Series . This sub-series focuses on high-quality content in the domain of mathematical sciences and various disciplines of mathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research, applied statistics and computer science. All content published in the sub-series are written, edited, or vetted by the laureates or jury members of the Infosys Prize. With this series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academic quality on current topics in relevant disciplines. Literature in this sub-series will appeal to a wide audience of researchers, students, educators, and professionals across mathematics, applied mathematics, statistics and computer science disciplines.The Infosys Science Foundation Series is a collaborative effort between the Infosys Science Foundation and Springer to publish high quality content. All content published in the Series is written, edited, or vetted by the laureates or jury members of the Infosys Prize. This umbrella series publishes content across academic domains, including, but not limited to, Applied Sciences, Engineering, Pure Sciences, and Mathematics. With this Series, Springer and the Infosys Science Foundation hope to provide readers with textbooks, monographs, handbooks, and professional books of the highest academic quality. Literature in this Series will appeal to a wide audience of researchers, students, educators, and professionals across disciplines.
More information about this subseries at http://​www.​springer.​com/​series/​13817

Tarlok Nath Shorey

Complex Analysis with Applications to Number Theory 1st ed. 2020

Tarlok Nath Shorey

Department of Natural Sciences and Engineering, National Institute of Advanced Studies, Bengaluru, Karnataka, India
ISSN 2363-6149 e-ISSN 2363-6157 Infosys Science Foundation Series
ISSN 2364-4036 e-ISSN 2364-4044 Infosys Science Foundation Series in Mathematical Sciences
ISBN 978-981-15-9096-2 e-ISBN 978-981-15-9097-9
https://doi.org/10.1007/978-981-15-9097-9
© Springer Nature Singapore Pte Ltd. 2020
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To Our Grandson Aarav

Preface

The text of the present book is based on my lectures at the School of Mathematics, Tata Institute of Fundamental Research (TIFR), Mumbai, India, during 1976–1977, at the Department of Mathematics, Punjab University, India, during 1977–1979 and at the Department of Mathematics, Indian Institute of Technology Bombay, India, during 2011–2016. I am indebted to my colleague Late Prof. R. R. Simha for discussions during my above lectures on Complex Analysis at TIFR. The book is divided into two parts. The first part (Chaps. 1 – 6 ) is on Complex Analysis, and the second part (Chaps. 7 – 10 ) covers some of the topics in Number Theory, where the theory on Complex Analysis included in the first part finds its relevance and applications.
In Chap. 1 , basic notions like connectedness, extended complex plane, complex integral, winding number, homotopic paths and simply connected regions are introduced. It is shown in Chap. 1 that the winding number at a point along two closed homotopic paths is equal, and the complement of a simply connected region in the extended complex plane is connected. The former statement is extended in Chap. 2 by proving that the integrals of an analytic function in a simply connected region along two closed homotopic paths are equal, and this implies the Cauchy theorem for closed paths in a simply connected region. This is also extended by a different method to the cycles homologous to zero in an open set in Chap. 2 where well-known theorems like Maximum modulus principle, Open mapping theorem, Inverse function theorem, Rouché theorem and Jensen inequality are derived from the Cauchy theorems. Regarding the latter statement that a complement of a simply connected region is connected, its converse is also valid. In fact, the Riemann mapping theorem is proved in Chap. 3 , and it implies that a region is simply connected if and only if its complement in the extended plane is connected. Groups of automorphisms of the open unit disc and the upper half plane are also determined in Chap. 3 . Harmonic functions are introduced in Chap. 4 where it is proved that a region is simply connected if and only if every harmonic function in has a harmonic conjugate in The Weierstrass factorisation theorem and the Hadamard factorisation theorem are proved in Chap. 6 leading to the gamma function and the Stirling formula.
The second part begins with the Riemann Zeta function . We prove in Chap. 7 that has no zero on the line . Further, we show that the Prime Number Theorem is equivalent to the non-vanishing of on the line by proving the Wiener-Ikehara theorem. Thus, a complete proof of the Prime Number Theorem has been given in this chapter. We give another proof in the next chapter by following the original and classical method of Hadamard and de la Vallée Poussin. In fact, we prove a stronger version with an error term in Chap. 8 . Further, we give in Chap. 7 an analytic continuation of in and prove its functional equation. Further, we show that where runs through the non-trivial zeros of and give an account of well-known conjectures in the theory of ; in particular, we show that the Riemann hypothesis implies Lindelöf hypothesis. The proofs of both the results depend on the Borel-Carathéodry lemma proved in Chap. 5 , where it has been applied for a proof of the Little Picard theorem and the Great Picard theorem. In Chap. 9 , we consider the Dirichlet series. An important class of the Dirichlet series is the Dirichlet -function where is a Dirichlet character. We prove their non-vanishing at for all the Dirichlet characters and derive the Dirichlet theorem that there are infinitely many primes in an arithmetic progression. In the last Chap. 10 , we prove the Baker theorem that the linear independence of logarithms of algebraic numbers over rationals implies their linear independence over algebraic numbers. The proof depends on Cauchy residue theorem and an estimate for the number of zeros of an exponential polynomial in a disc proved in Chap. 2 justifying to include Chap. 10 in this book. A list of exercises is given at the end of each chapter; hints are provided for some of them, and some chapters contain examples with complete solutions.
This book has 13 figures. I thank Bidisha (HRI) for drawing these figures and Bidisha, Divyum (BITS-Pillani) and Saranya (IIITG) for their remarks on a draft of the book. I thank Bidisha, Sandhya (NIAS), Sneh (IISc) and Veekesh (IMSc) for typing the manuscript and Saranya for her interest in this project from the beginning. There was no type-setting in the manuscript, and R. Thangadurai has kindly agreed to undertake this essential, difficult and time-consuming task to bring the manuscript to the present state. Further, he carried out the never ending job of changes and corrections for finalising the draft of the book. I am indebted to his generous contributions. Further, I thank R. Tijdeman, T. N. Venkataramana and Michel Waldschmidt for their valuable remarks and suggestions. I am indebted to Late Professor Baldev Raj, Director NIAS, for his interest in this project; NIAS for excellent facilities and INSA for financial support. Further, I thank the referees for their useful remarks. Finally, I thank my wife Savita for her support when I was working on this book.
Tarlok Nath Shorey
Bangalore, India

Symbols

The complement of set in set
Distance between sets and , Exercise 1.20
Closure of
Interior of
Boundary of
The range of curve
Length of a curve , p. 10
open set
Open disc with centre and radius
Closed disc with cen