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**An invitation to modern number theory. With a foreword by Peter Sarnak.**
*(English)*
Zbl 1155.11001

Princeton, NJ: Princeton University Press (ISBN 0-691-12060-9/hbk). xx, 503 p. (2006).

From the text: “The beginning student of physics, chemistry, or computer science learns early on that in order to gain a proper understanding of the subject, one has to understand seemingly different topics and their relation to one another. While this is equally true in mathematics, this feature is not brought to the fore in most modern texts of pure mathematics. Different fields are usually presented as complete and isolated topics, and for the most part this is how it should be. However, modern mathematics, abstract as it might appear to the beginner, is driven by concrete and basic problems. In fact many of the different areas were developed in attempts, sometimes successful, to resolve such fundamental questions. Hence it should not be surprising that often the solution to a concrete long-standing problem involves combining different areas. This is especially true of modern number theory. The formulation of the problems is mostly elementary, and the expected truths were many times discovered by numerical experimentation.

In this lovely book the authors introduce number theory in terms of its connections to other fields of mathematics and its applications. While adhering to this theme, they also emphasize concrete problems (solved and unsolved). They develop the requisite mathematical background along the way to ensure proper and clear treatment of each of the many topics discussed. This allows the beginner to get an immediate taste of modern mathematics as well as of mathematical research. Naturally the treatments of various theories and theorems cannot be as complete as in books which are devoted to a single topic; however, through the indicated further reading, the many excellent exercises and the proposed research projects, the reader will get an excellent first understanding of the material.

Parts of this book have been used very successfully in undergraduate courses at the junior level at Princeton, New York University, Ohio State, and Brown. It will no doubt find similar success much more broadly, and it should appeal to both more- and less-advanced students. Covering most of the material in this book is a challenging task for both the student (or reader) and the instructor. My experience in co-teaching (with one of the authors) a version of part of the material is that this effort results in rich rewards for both the student and the instructor. ” (Preface by Peter Sarnak)

In a manner accessible to beginning undergraduates, this textbook introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth’s Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

The authors introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach’s Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

Contents: Part 1. Basic Number Theory: Mod \(p\) Arithmetic, Group Theory and Cryptography; Arithmetic Functions; Zeta and \(L\)-Functions; Solutions to Diophantine Equations.

Part 2. Continued Fractions and Approximations: Algebraic and Transcendental Numbers; The Proof of Roth’s Theorem; Introduction to Continued Fractions.

Part 3. Probabilistic Methods and Equidistribution: Introduction to Probability; Applications of Probability: Benford’s Law and Hypothesis Testing; Distribution of Digits of Continued Fractions; Introduction to Fourier Analysis; \(\{n^k\alpha\}\) and Poissonian Behavior.

Part 4. The Circle Method: Introduction to the Circle Method; Circle Method: Heuristics for Germain Primes.

Part 5. Random Matrix Theory and \(L\)-Functions: From Nuclear Physics to \(L\)-Functions; Random Matrix Theory: Eigenvalue Densities; Random Matrix Theory: Spacings between Adjacent Eigenvalues; The Explicit Formula and Density Conjectures.

Appendixes A–D. Analysis Review; Linear Algebra Review; Hints and Remarks on the Exercises; Concluding Remarks.

Providing exercises, references to the background literature, and Web links to previous student research projects, this text can be used to teach a research seminar or a lecture class.

In this lovely book the authors introduce number theory in terms of its connections to other fields of mathematics and its applications. While adhering to this theme, they also emphasize concrete problems (solved and unsolved). They develop the requisite mathematical background along the way to ensure proper and clear treatment of each of the many topics discussed. This allows the beginner to get an immediate taste of modern mathematics as well as of mathematical research. Naturally the treatments of various theories and theorems cannot be as complete as in books which are devoted to a single topic; however, through the indicated further reading, the many excellent exercises and the proposed research projects, the reader will get an excellent first understanding of the material.

Parts of this book have been used very successfully in undergraduate courses at the junior level at Princeton, New York University, Ohio State, and Brown. It will no doubt find similar success much more broadly, and it should appeal to both more- and less-advanced students. Covering most of the material in this book is a challenging task for both the student (or reader) and the instructor. My experience in co-teaching (with one of the authors) a version of part of the material is that this effort results in rich rewards for both the student and the instructor. ” (Preface by Peter Sarnak)

In a manner accessible to beginning undergraduates, this textbook introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth’s Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

The authors introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach’s Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

Contents: Part 1. Basic Number Theory: Mod \(p\) Arithmetic, Group Theory and Cryptography; Arithmetic Functions; Zeta and \(L\)-Functions; Solutions to Diophantine Equations.

Part 2. Continued Fractions and Approximations: Algebraic and Transcendental Numbers; The Proof of Roth’s Theorem; Introduction to Continued Fractions.

Part 3. Probabilistic Methods and Equidistribution: Introduction to Probability; Applications of Probability: Benford’s Law and Hypothesis Testing; Distribution of Digits of Continued Fractions; Introduction to Fourier Analysis; \(\{n^k\alpha\}\) and Poissonian Behavior.

Part 4. The Circle Method: Introduction to the Circle Method; Circle Method: Heuristics for Germain Primes.

Part 5. Random Matrix Theory and \(L\)-Functions: From Nuclear Physics to \(L\)-Functions; Random Matrix Theory: Eigenvalue Densities; Random Matrix Theory: Spacings between Adjacent Eigenvalues; The Explicit Formula and Density Conjectures.

Appendixes A–D. Analysis Review; Linear Algebra Review; Hints and Remarks on the Exercises; Concluding Remarks.

Providing exercises, references to the background literature, and Web links to previous student research projects, this text can be used to teach a research seminar or a lecture class.

Reviewer: Olaf Ninnemann (Berlin)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11A25 | Arithmetic functions; related numbers; inversion formulas |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |

11Dxx | Diophantine equations |

11Jxx | Diophantine approximation, transcendental number theory |

11Kxx | Probabilistic theory: distribution modulo \(1\); metric theory of algorithms |

11P32 | Goldbach-type theorems; other additive questions involving primes |

11P55 | Applications of the Hardy-Littlewood method |

15B52 | Random matrices (algebraic aspects) |

11Z05 | Miscellaneous applications of number theory |