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| Softcover ISBN: | 978-1-4704-1851-9 |
| Product Code: | ULECT/62 |
| List Price: | $69.00 |
| MAA Member Price: | $62.10 |
| AMS Member Price: | $55.20 |
| eBook ISBN: | 978-1-4704-1891-5 |
| Product Code: | ULECT/62.E |
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| AMS Member Price: | $52.00 |
| Softcover ISBN: | 978-1-4704-1851-9 |
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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 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.ReadershipUndergraduate and graduate students and research mathematicians interested in number theory and complex analysis.
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Table of Contents
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Part 1. Classical topics
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Chapter 1. Panorama of arithmetic functions
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Chapter 2. The Euler–Maclaurin formula
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Chapter 3. Tchebyshev’s prime seeds
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Chapter 4. Elementary prime number theorem
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Chapter 5. The Riemann memoir
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Chapter 6. The analytic continuation
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Chapter 7. The functional equation
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Chapter 8. The product formula over the zeros
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Chapter 9. The asymptotic formula for $N(T)$
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Chapter 10. The asymptotic formula for $\psi (x)$
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Chapter 11. The zero-free region and the PNT
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Chapter 12. Approximate functional equations
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Chapter 13. The Dirichlet polynomials
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Chapter 14. Zeros off the critical line
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Chapter 15. Zeros on the critical line
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Part 2. The critical zeros after Levinson
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Chapter 16. Introduction
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Chapter 17. Detecting critical zeros
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Chapter 18. Conrey’s construction
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Chapter 19. The argument variations
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Chapter 20. Attaching a mollifier
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Chapter 21. The Littlewood lemma
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Chapter 22. The principal inequality
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Chapter 23. Positive proportion of the critical zeros
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Chapter 24. The first moment of Dirichlet polynomials
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Chapter 25. The second moment of Dirichlet polynomials
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Chapter 26. The diagonal terms
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Chapter 27. The off-diagonal terms
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Chapter 28. Conclusion
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Chapter 29. Computations and the optimal mollifier
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Appendix A. Smooth bump functions
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Appendix B. The gamma function
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Additional Material
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Reviews
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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
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- Book Details
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- Reviews
- Requests
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.
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 zero-free 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 off-diagonal 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, 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
