**University Lecture Series**

Volume: 62;
2014;
119 pp;
Softcover

MSC: Primary 11;

Print ISBN: 978-1-4704-1851-9

Product Code: ULECT/62

List Price: $40.00

Individual Member Price: $32.00

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**Electronic ISBN: 978-1-4704-1891-5
Product Code: ULECT/62.E**

List Price: $40.00

Individual Member Price: $32.00

#### Supplemental Materials

# Lectures on the Riemann Zeta Function

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*H. Iwaniec*

The Riemann zeta function was introduced by
L. Euler (1737) in connection with questions about the distribution of
prime numbers. Later, B. Riemann (1859) derived deeper results about
the prime numbers by considering the zeta function in the complex
variable. The famous Riemann Hypothesis, asserting that all of the
non-trivial zeros of zeta are on a critical line in the complex plane,
is one of the most important unsolved problems in modern mathematics.

The present book consists of two parts. The first part covers
classical material about the zeros of the Riemann zeta function with
applications to the distribution of prime numbers, including those
made by Riemann himself, F. Carlson, and Hardy–Littlewood. The second
part gives a complete presentation of Levinson's method for zeros on
the critical line, which allows one to prove, in particular, that more
than one-third of non-trivial zeros of zeta are on the critical
line. This approach and some results concerning integrals of Dirichlet
polynomials are new. There are also technical lemmas which can be
useful in a broader context.

#### Table of Contents

# Table of Contents

## Lectures on the Riemann Zeta Function

- Cover Cover11 free
- Title page iii4 free
- Table of contents v6 free
- Preface vii8 free
- Part 1. Classical Topics 110 free
- 1. Panorama of Arithmetic Functions 312
- 2. The Euler–Maclaurin Formula 918
- 3. Tchebyshev's Prime Seeds 1322
- 4. Elementary Prime Number Theorem 1524
- 5. The Riemann Memoir 2130
- 6. The Analytic Continuation 2332
- 7. The Functional Equation 2534
- 8. The Product Formula over the Zeros 2736
- 9. The Asymptotic Formula for N(T) 3342
- 10. The Asymptotic Formula for Ψ(x) 3746
- 11. The Zero-free Region and the PNT 4150
- 12. Approximate Functional Equations 4352
- 13. The Dirichlet Polynomials 4756
- 14. Zeros off the Critical Line 5564
- 15. Zeros on the Critical Line 5766
- Part 2. The Critical Zeros after Levinson 6372
- 16. Introduction 6574
- 17. Detecting Critical Zeros 6776
- 18. Conrey's Construction 6978
- 19. The Argument Variations 7180
- 20. Attaching a Mollifier 7584
- 21. The Littlewood Lemma 7786
- 22. The Principal Inequality 7988
- 23. Positive Proportion of the Critical Zeros 8190
- 24. The First Moment of Dirichlet Polynomials 8392
- 25. The Second Moment of Dirichlet Polynomials 8594
- 26. The Diagonal Terms 8796
- 27. The Off-diagonal Terms 95104
- 28. Conclusion 103112
- 29. Computations and the Optimal Mollifier 107116
- Appendix A. Smooth Bump Functions 111120
- Appendix B. The Gamma Function 115124
- Bibliography 117126
- Index 119128 free
- Back Cover Back Cover1130

#### Readership

Undergraduate and graduate students and research mathematicians interested in number theory and complex analysis.

#### Reviews

Amazingly, this slim book will take you from the basics to the frontiers on Riemann's zeta function.

-- Tamás Waldhauser, ACTA Sci. Math.

The book under review presents a number of essential research directions in this area in a friendly fashion. It focuses mainly on two questions: the location of zeros in the critical strip (the strip on the complex plane consisting of numbers with real part between 0 and 1) and the proportion of zeros on the critical line (the line at the center of the strip, consisting of numbers with real part 1/2). ... The book is quite technical, and readers need a basic knowledge in complex function theory and also analytic number theory to follow the details. The chapters are short, well-motivated, and well written; there are several exercises. Thus, the book can serve as a source for researchers working on the Riemann zeta-function and also to be a good text for an advanced graduate course.

-- MAA Reviews