Softcover ISBN:  9781470418519 
Product Code:  ULECT/62 
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AMS Member Price:  $34.40 
Electronic ISBN:  9781470418915 
Product Code:  ULECT/62.E 
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Book DetailsUniversity Lecture SeriesVolume: 62; 2014; 119 ppMSC: Primary 11;
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 nontrivial 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 onethird of nontrivial 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.ReadershipUndergraduate and graduate students and research mathematicians interested in number theory and complex analysis.

Table of Contents

Part 1. Classical topics

Chapter 1. Panorama of arithmetic functions

Chapter 2. The Euler–Maclaurin formula

Chapter 3. Tchebyshev’s prime seeds

Chapter 4. Elementary prime number theorem

Chapter 5. The Riemann memoir

Chapter 6. The analytic continuation

Chapter 7. The functional equation

Chapter 8. The product formula over the zeros

Chapter 9. The asymptotic formula for $N(T)$

Chapter 10. The asymptotic formula for $\psi (x)$

Chapter 11. The zerofree region and the PNT

Chapter 12. Approximate functional equations

Chapter 13. The Dirichlet polynomials

Chapter 14. Zeros off the critical line

Chapter 15. Zeros on the critical line

Part 2. The critical zeros after Levinson

Chapter 16. Introduction

Chapter 17. Detecting critical zeros

Chapter 18. Conrey’s construction

Chapter 19. The argument variations

Chapter 20. Attaching a mollifier

Chapter 21. The Littlewood lemma

Chapter 22. The principal inequality

Chapter 23. Positive proportion of the critical zeros

Chapter 24. The first moment of Dirichlet polynomials

Chapter 25. The second moment of Dirichlet polynomials

Chapter 26. The diagonal terms

Chapter 27. The offdiagonal terms

Chapter 28. Conclusion

Chapter 29. Computations and the optimal mollifier

Appendix A. Smooth bump functions

Appendix B. The gamma function


Additional Material

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, wellmotivated, and well written; there are several exercises. Thus, the book can serve as a source for researchers working on the Riemann zetafunction and also to be a good text for an advanced graduate course.
MAA Reviews


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

Part 1. Classical topics

Chapter 1. Panorama of arithmetic functions

Chapter 2. The Euler–Maclaurin formula

Chapter 3. Tchebyshev’s prime seeds

Chapter 4. Elementary prime number theorem

Chapter 5. The Riemann memoir

Chapter 6. The analytic continuation

Chapter 7. The functional equation

Chapter 8. The product formula over the zeros

Chapter 9. The asymptotic formula for $N(T)$

Chapter 10. The asymptotic formula for $\psi (x)$

Chapter 11. The zerofree region and the PNT

Chapter 12. Approximate functional equations

Chapter 13. The Dirichlet polynomials

Chapter 14. Zeros off the critical line

Chapter 15. Zeros on the critical line

Part 2. The critical zeros after Levinson

Chapter 16. Introduction

Chapter 17. Detecting critical zeros

Chapter 18. Conrey’s construction

Chapter 19. The argument variations

Chapter 20. Attaching a mollifier

Chapter 21. The Littlewood lemma

Chapter 22. The principal inequality

Chapter 23. Positive proportion of the critical zeros

Chapter 24. The first moment of Dirichlet polynomials

Chapter 25. The second moment of Dirichlet polynomials

Chapter 26. The diagonal terms

Chapter 27. The offdiagonal terms

Chapter 28. Conclusion

Chapter 29. Computations and the optimal mollifier

Appendix A. Smooth bump functions

Appendix B. The gamma function

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, wellmotivated, and well written; there are several exercises. Thus, the book can serve as a source for researchers working on the Riemann zetafunction and also to be a good text for an advanced graduate course.
MAA Reviews