This volume contains a collection of research and survey papers written by some of the most eminent mathematicians in the international community and is dedicated to Helmut Maier, whose own research has been groundbreaking and deeply influential to the field. Specific emphasis is given to topics regarding exponential and trigonometric sums and their behavior in short intervals, anatomy of integers and cyclotomic polynomials, small gaps in sequences of sifted prime numbers, oscillation theorems for primes in arithmetic progressions, inequalities related to the distribution of primes in short intervals, the Moebius function, Euler's totient function, the Riemann zeta function and the Riemann Hypothesis. Graduate students, research mathematicians, as well as computer scientists and engineers who are interested in pure and interdisciplinary research, will find this volume a useful resource. Contributors to this volume: Bill Allombert, Levent Alpoge, Nadine Amersi, Yuri Bilu, Regis de la Breteche, Christian Elsholtz, John B. Friedlander, Kevin Ford, Daniel A. Goldston, Steven M. Gonek, Andrew Granville, Adam J. Harper, Glyn Harman, D. R. Heath-Brown, Aleksandar Ivic, Geoffrey Iyer, Jerzy Kaczorowski, Daniel M. Kane, Sergei Konyagin, Dimitris Koukoulopoulos, Michel L. Lapidus, Oleg Lazarev, Andrew H. Ledoan, Robert J. Lemke Oliver, Florian Luca, James Maynard, Steven J. Miller, Hugh L. Montgomery, Melvyn B. Nathanson, Ashkan Nikeghbali, Alberto Perelli, Amalia Pizarro-Madariaga, Janos Pintz, Paul Pollack, Carl Pomerance, Michael Th. Rassias, Maksym Radziwill, Joel Rivat, Andras Sarkoezy, Jeffrey Shallit, Terence Tao, Gerald Tenenbaum, Laszlo Toth, Tamar Ziegler, Liyang Zhang.

This book focuses on complex geometry and covers highly active topics centered around geometric problems in several complex variables and complex dynamics, written by some of the world's leading experts in their respective fields. This book features research and expository contributions from the 2013 Abel Symposium, held at the Norwegian University of Science and Technology Trondheim on July 2-5, 2013. The purpose of the symposium was to present the state of the art on the topics, and to discuss future research directions.

Covering topics in graph theory, L-functions, p-adic geometry, Galois representations, elliptic fibrations, genus 3 curves and bad reduction, harmonic analysis, symplectic groups and mould combinatorics, this volume presents a collection of papers covering a wide swath of number theory emerging from the third iteration of the international Women in Numbers conference, "Women in Numbers - Europe" (WINE), held on October 14-18, 2013 at the CIRM-Luminy mathematical conference center in France. While containing contributions covering a wide range of cutting-edge topics in number theory, the volume emphasizes those concrete approaches that make it possible for graduate students and postdocs to begin work immediately on research problems even in highly complex subjects.

Peter Higgins distills centuries of work into one delightful narrative that celebrates the mystery of numbers and explains how different kinds of numbers arose and why they are useful. Full of historical snippets and interesting examples, the book ranges from simple number puzzles and magic tricks, to showing how ideas about numbers relate to real-world problems. This fascinating book will inspire and entertain readers across a range of abilities. Easy material is blended with more challenging ideas. As our understanding of numbers continues to evolve, this book invites us to rediscover the mystery and beauty of numbers.

The aim of this book is to provide a comprehensive account of higher dimensional Nevanlinna theory and its relations with Diophantine approximation theory for graduate students and interested researchers. This book with nine chapters systematically describes Nevanlinna theory of meromorphic maps between algebraic varieties or complex spaces, building up from the classical theory of meromorphic functions on the complex plane with full proofs in Chap. 1 to the current state of research. Chapter 2 presents the First Main Theorem for coherent ideal sheaves in a very general form. With the preparation of plurisubharmonic functions, how the theory to be generalized in a higher dimension is described. In Chap. 3 the Second Main Theorem for differentiably non-degenerate meromorphic maps by Griffiths and others is proved as a prototype of higher dimensional Nevanlinna theory. Establishing such a Second Main Theorem for entire curves in general complex algebraic varieties is a wide-open problem. In Chap. 4, the Cartan-Nochka Second Main Theorem in the linear projective case and the Logarithmic Bloch-Ochiai Theorem in the case of general algebraic varieties are proved. Then the theory of entire curves in semi-abelian varieties, including the Second Main Theorem of Noguchi-Winkelmann-Yamanoi, is dealt with in full details in Chap. 6. For that purpose Chap. 5 is devoted to the notion of semi-abelian varieties. The result leads to a number of applications. With these results, the Kobayashi hyperbolicity problems are discussed in Chap. 7. In the last two chapters Diophantine approximation theory is dealt with from the viewpoint of higher dimensional Nevanlinna theory, and the Lang-Vojta conjecture is confirmed in some cases. In Chap. 8 the theory over function fields is discussed. Finally, in Chap. 9, the theorems of Roth, Schmidt, Faltings, and Vojta over number fields are presented and formulated in view of Nevanlinna theory with results motivated by those in Chaps. 4, 6, and 7.

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie-Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.

This textbook explores a selection of topics in complex analysis. From core material in the mainstream of complex analysis itself, to tools that are widely used in other areas of mathematics, this versatile compilation offers a selection of many different paths. Readers interested in complex analysis will appreciate the unique combination of topics and connections collected in this book. Beginning with a review of the main tools of complex analysis, harmonic analysis, and functional analysis, the authors go on to present multiple different, self-contained avenues to proceed. Chapters on linear fractional transformations, harmonic functions, and elliptic functions offer pathways to hyperbolic geometry, automorphic functions, and an intuitive introduction to the Schwarzian derivative. The gamma, beta, and zeta functions lead into L-functions, while a chapter on entire functions opens pathways to the Riemann hypothesis and Nevanlinna theory. Cauchy transforms give rise to Hilbert and Fourier transforms, with an emphasis on the connection to complex analysis. Valuable additional topics include Riemann surfaces, steepest descent, tauberian theorems, and the Wiener-Hopf method. Showcasing an array of accessible excursions, Explorations in Complex Functions is an ideal companion for graduate students and researchers in analysis and number theory. Instructors will appreciate the many options for constructing a second course in complex analysis that builds on a first course prerequisite; exercises complement the results throughout.

This textbook provides an accessible account of the history of abstract algebra, tracing a range of topics in modern algebra and number theory back to their modest presence in the seventeenth and eighteenth centuries, and exploring the impact of ideas on the development of the subject. Beginning with Gauss's theory of numbers and Galois's ideas, the book progresses to Dedekind and Kronecker, Jordan and Klein, Steinitz, Hilbert, and Emmy Noether. Approaching mathematical topics from a historical perspective, the author explores quadratic forms, quadratic reciprocity, Fermat's Last Theorem, cyclotomy, quintic equations, Galois theory, commutative rings, abstract fields, ideal theory, invariant theory, and group theory. Readers will learn what Galois accomplished, how difficult the proofs of his theorems were, and how important Camille Jordan and Felix Klein were in the eventual acceptance of Galois's approach to the solution of equations. The book also describes the relationship between Kummer's ideal numbers and Dedekind's ideals, and discusses why Dedekind felt his solution to the divisor problem was better than Kummer's. Designed for a course in the history of modern algebra, this book is aimed at undergraduate students with an introductory background in algebra but will also appeal to researchers with a general interest in the topic. With exercises at the end of each chapter and appendices providing material difficult to find elsewhere, this book is self-contained and therefore suitable for self-study.

In 2013, a little known mathematician in his late 50s stunned the mathematical community with a breakthrough on an age-old problem about prime numbers. Since then, there has been further dramatic progress on the problem, thanks to the efforts of a large-scale online collaborative effort of a type that would have been unthinkable in mathematics a couple of decades ago, and the insight and creativity of a young mathematician at the start of his career. Prime numbers have intrigued, inspired and infuriated mathematicians for millennia. Every school student studies prime numbers and can appreciate their beauty, and yet mathematicians' difficulty with answering some seemingly simple questions about them reveals the depth and subtlety of prime numbers. Vicky Neale charts the recent progress towards proving the famous Twin Primes Conjecture, and the very different ways in which the breakthroughs have been made: a solo mathematician working in isolation and obscurity, and a large collaboration that is more public than any previous collaborative effort in mathematics and that reveals much about how mathematicians go about their work. Interleaved with this story are highlights from a significantly older tale, going back two thousand years and more, of mathematicians' efforts to comprehend the beauty and unlock the mysteries of the prime numbers.

Number Theory Revealed: A Masterclass acquaints enthusiastic students with the ""Queen of Mathematics''. The text offers a fresh take on congruences, power residues, quadratic residues, primes, and Diophantine equations and presents hot topics like cryptography, factoring, and primality testing. Students are also introduced to beautiful enlightening questions like the structure of Pascal's triangle mod $p$ and modern twists on traditional questions like the values represented by binary quadratic forms, the anatomy of integers, and elliptic curves. This Masterclass edition contains many additional chapters and appendices not found in Number Theory Revealed: An Introduction, highlighting beautiful developments and inspiring other subjects in mathematics (like algebra). This allows instructors to tailor a course suited to their own (and their students') interests. There are new yet accessible topics like the curvature of circles in a tiling of a circle by circles, the latest discoveries on gaps between primes, a new proof of Mordell's Theorem for congruent elliptic curves, and a discussion of the $abc$-conjecture including its proof for polynomials.

In its simplest form, Hodge theory is the study of periods - integrals of algebraic differential forms which arise in the study of complex geometry and moduli, number theory and physics. Organized around the basic concepts of variations of Hodge structure and period maps, this volume draws together new developments in deformation theory, mirror symmetry, Galois representations, iterated integrals, algebraic cycles and the Hodge conjecture. Its mixture of high-quality expository and research articles make it a useful resource for graduate students and seasoned researchers alike.

The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry. It is also essential reading for any researcher wishing to keep abreast of the latest developments in the field. Highlights include Tim Browning's survey on applications of the circle method to rational points on algebraic varieties and Per Salberger's chapter on rational points on cubic hypersurfaces.

L-functions associated to automorphic forms encode all classical number theoretic information. They are akin to elementary particles in physics. This book provides an entirely self-contained introduction to the theory of L-functions in a style accessible to graduate students with a basic knowledge of classical analysis, complex variable theory, and algebra. Also within the volume are many new results not yet found in the literature. The exposition provides complete detailed proofs of results in an easy-to-read format using many examples and without the need to know and remember many complex definitions. The main themes of the book are first worked out for GL(2,R) and GL(3,R), and then for the general case of GL(n,R). In an appendix to the book, a set of Mathematica functions is presented, designed to allow the reader to explore the theory from a computational point of view.

This book studies the interplay between the geometry and topology of locally symmetric spaces, and the arithmetic aspects of the special values of L-functions. The authors study the cohomology of locally symmetric spaces for GL(N) where the cohomology groups are with coefficients in a local system attached to a finite-dimensional algebraic representation of GL(N). The image of the global cohomology in the cohomology of the Borel-Serre boundary is called Eisenstein cohomology, since at a transcendental level the cohomology classes may be described in terms of Eisenstein series and induced representations. However, because the groups are sheaf-theoretically defined, one can control their rationality and even integrality properties. A celebrated theorem by Langlands describes the constant term of an Eisenstein series in terms of automorphic L-functions. A cohomological interpretation of this theorem in terms of maps in Eisenstein cohomology allows the authors to study the rationality properties of the special values of Rankin-Selberg L-functions for GL(n) x GL(m), where n + m = N. The authors carry through the entire program with an eye toward generalizations. This book should be of interest to advanced graduate students and researchers interested in number theory, automorphic forms, representation theory, and the cohomology of arithmetic groups.

This book offers an original contribution to the foundations of logic and mathematics and focuses on the internal logic of mathematical theories, from arithmetic or number theory to algebraic geometry. Arithmetical logic is the term used to refer to the internal logic of classical arithmetic, here called Fermat-Kronecker arithmetic and combines Fermat's method of infinite descent with Kronecker's general arithmetic of homogeneous polynomials. The book also includes a treatment of theories in physics and mathematical physics to underscore the role of arithmetic from a constructivist viewpoint. The scope of the work intertwines historical, mathematical, logical and philosophical dimensions in a unified critical perspective; as such, it will appeal to a broad readership from mathematicians to logicians, to philosophers interested in foundational questions. Researchers and graduate students in the fields of philosophy and mathematics will benefit from the author's critical approach to the foundations of logic and mathematics.

In the last decade, the areas of quadratic and higher degree forms have witnessed dramatic advances. This volume is an outgrowth of three seminal conferences on these topics held in 2009, two at the University of Florida and one at the Arizona Winter School. The volume also includes papers from the two focused weeks on quadratic forms and integral lattices at the University of Florida in 2010.Topics discussed include the links between quadratic forms and automorphic forms, representation of integers and forms by quadratic forms, connections between quadratic forms and lattices, and algorithms for quaternion algebras and quadratic forms. The book will be of interest to graduate students and mathematicians wishing to study quadratic and higher degree forms, as well as to established researchers in these areas. Quadratic and Higher Degree Forms contains research and semi-expository papers that stem from the presentations at conferences at the University of Florida as well as survey lectures on quadratic forms based on the instructional workshop for graduate students held at the Arizona Winter School. The survey papers in the volume provide an excellent introduction to various aspects of the theory of quadratic forms starting from the basic concepts and provide a glimpse of some of the exciting questions currently being investigated. The research and expository papers present the latest advances on quadratic and higher degree forms and their connections with various branches of mathematics.

This volume, as Andrew M. Odlzyko writes in the foreword, "commemorates and celebrates the life and achievements of an extraordinary person." Originally conceived as an 80th birthday tribute to Herbert Wilf, the well-known combinatorialist, the book has evolved beyond the proceeds of the W80 tribute. Professor Wilf was an award-winning teacher, who was supportive of women mathematicians, and who had an unusually high proportion of women among his PhD candidates. He was Editor-in-chief of the American Mathematical Monthly and a founder of both the Journal of Algorithms and of the Electronic Journal of Combinatorics. But he was first a researcher, driven by his desire to know and explain the inner workings of the mathematical world. The book collects high-quality, refereed research contributions by some of Professor Wilf's colleagues, students, and collaborators. Many of the papers presented here were featured in the Third Waterloo Workshop on Computer Algebra (WWCA 2011, W80), held May 26-29, 2011 at Wilfrid Laurier University, Waterloo, Canada. Others were included because of their relationship to his important work in combinatorics. All are presented as a tribute to Herb Wilf's contributions to mathematics and mathematical life.

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch-Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

George Andrews is one of the most influential figures in number theory and combinatorics. In the theory of partitions and q-hypergeometric series and in the study of Ramanujan's work, he is the unquestioned leader. To suitably honor him during his 70th birthday year, an International Conference on Combinatory Analysis was held at The Pennsylvania State University during December 5-7, 2008. Three issues of the Ramanujan Journal comprising Volume 23 were published in 2010 as the refereed proceedings of that conference. The Ramanujan Journal was proud to bring out that volume honoring one of its Founding Editors. In view of the great interest that the mathematical community has in the influential work of Andrews, it was decided to republish Volume 23 of The Ramanujan Journal in this book form, so that the refereed proceedings are more readily available for those who do not subscribe to the journal but wish to possess this volume. As a fitting tribute to George Andrews, many speakers from the conference contributed research papers to this volume which deals with a broad range of areas that signify the research interests of George Andrews. In reproducing Volume 23 of The Ramanujan Journal in this book form, we have included two papers-one by Hei-Chi Chan and Shaun Cooper, and another by Ole Warnaar-which were intended for Volume 23 of The Ramanujan Journal, but appeared in other issues. The enormous productivity of George Andrews remains unabated in spite of the passage of time. His immensely fertile mind continues to pour forth seminal ideas year after year. He has two research papers in this volume. May his eternal youthfulness and his magnificent research output continue to inspire and influence researchers in the years ahead.

This is the most comprehensive survey of the mathematical life of the legendary Paul Erdos (1913-1996), one of the most versatile and prolific mathematicians of our time. For the first time, all the main areas of Erdos' research are covered in a single project. Because of overwhelming response from the mathematical community, the project now occupies over 1000 pages, arranged into two volumes. These volumes contain both high level research articles as well as key articles that survey some of the cornerstones of Erdos' work, each written by a leading world specialist in the field. A special chapter "Early Days", rare photographs, and art related to Erdos complement this striking collection. A unique contribution is the bibliography on Erdos' publications: the most comprehensive ever published. This new edition, dedicated to the 100th anniversary of Paul Erdos' birth, contains updates on many of the articles from the two volumes of the first edition, several new articles from prominent mathematicians, a new introduction, more biographical information about Paul Erdos, and an updated list of publications. The first volume contains the unique chapter "Early Days", which features personal memories of Paul Erdos by a number of his colleagues. The other three chapters cover number theory, random methods, and geometry. All of these chapters are essentially updated, most notably the geometry chapter that covers the recent solution of the problem on the number of distinct distances in finite planar sets, which was the most popular of Erdos' favorite geometry problems.

Pseudo-reductive groups arise naturally in the study of general smooth linear algebraic groups over non-perfect fields and have many important applications. This monograph provides a comprehensive treatment of the theory of pseudo-reductive groups and gives their classification in a usable form. In this second edition there is new material on relative root systems and Tits systems for general smooth affine groups, including the extension to quasi-reductive groups of famous simplicity results of Tits in the semisimple case. Chapter 9 has been completely rewritten to describe and classify pseudo-split absolutely pseudo-simple groups with a non-reduced root system over arbitrary fields of characteristic 2 via the useful new notion of 'minimal type' for pseudo-reductive groups. Researchers and graduate students working in related areas, such as algebraic geometry, algebraic group theory, or number theory will value this book, as it develops tools likely to be used in tackling other problems.

Automorphic forms and Galois representations have played a central role in the development of modern number theory, with the former coming to prominence via the celebrated Langlands program and Wiles' proof of Fermat's Last Theorem. This two-volume collection arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic Forms and Galois Representations' in July 2011, the aim of which was to explore recent developments in this area. The expository articles and research papers across the two volumes reflect recent interest in p-adic methods in number theory and representation theory, as well as recent progress on topics from anabelian geometry to p-adic Hodge theory and the Langlands program. The topics covered in volume one include the Shafarevich Conjecture, effective local Langlands correspondence, p-adic L-functions, the fundamental lemma, and other topics of contemporary interest.

The techniques presented here are useful for solving mathematical contest problems in algebra and analysis. Most of the examples and exercises that appear in the book originate from mathematical Olympiad competitions around the world. In the first four chapters the authors cover material for competitions at high school level. The level advances with the chapters. The topics explored include polynomials, functional equations, sequences and an elementary treatment of complex numbers. The final chapters provide a comprehensive list of problems posed at national and international contests in recent years, and solutions to all exercises and problems presented in the book. It helps students in preparing for national and international mathematical contests form high school level to more advanced competitions and will also be useful for their first year of mathematical studies at the university. It will be of interest to teachers in college and university level, and trainers of the mathematical Olympiads.

The book describes 4 main topics in current model theory and updates their most recent development and applications. The 4 topics are: 1) model theory of valued fields; 2) undecidability in arithmetic; 3) NIP theories; 4) model theory of real and complex exponentiation. The book addresses in particular young researchers in model theory, as well as more senior researchers in other branches of mathematics.

Originally published in 1915 as number eighteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, and here reissued in its 1952 reprinted form, this book contains a condensed account of Dirichlet's Series, which relates to number theory. This tract will be of value to anyone with an interest in the history of mathematics or in the work of G. H. Hardy.