Lectures on the Riemann zeta function / H. Iwaniec.
Material type:
TextSeries: University lecture series ; v 62.Publication details: Providence, Rhode Island : American Mathematical Society, 2014.Description: vii, 119 p. : illustrations ; 26 cmISBN: - 9781470418519 (alk. paper)
- Riemann zeta function
- 515.56 23 Iw96
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| Books | ISI Library, Kolkata | 515.56 Iw96 (Browse shelf(Opens below)) | Available | 136019 |
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| 515.56 H473 Selberg trace formula for PSL(2.IR) | 515.56 In61 Report | 515.56 Iv95 Lectures on mean values of the Riemann zeta function | 515.56 Iw96 Lectures on the Riemann zeta function / | 515.56 J93 Cohomological theory of dynamical Zeta functions | 515.56 K16 Contributions to the theory of zeta-functions : | 515.56 K19 Random matrices, Frobenius eigenvalues, and monodromy |
Includes bibliographical references (page 117) and index.
Part 1. Classical topics
1. Panorama of arithmetic functions--
2. The Euler-Maclaurin Formula--
3. Tchebyshev's prime seeds--
4. Elementary prime number theorem--
5. The Riemann memoir--
6. The analytic continuation--
7. The functional equation--
8. The product formula over the zeros--
9. The asymptotic formula for N(T)--
10. The asymptotic formula for ?(x)--
11. The zero-free region and the PNT--
12. Approximate functional equations--
13. The Dirichlet polynomials--
14. Zeros off the critical line--
Part 2. The critical zeros after Levinson:
16. Introduction--
17. Detecting critical zeros--
18. Conrey's construction--
19. The argument variations--
20. Attaching a mollifier--
21. The Littlewood lemma--
22. The principal inequality--
23. Positive proportion of the critical zeros--
24. The first moment of Dirichlet polynomials--
25. The second moment of Dirichlet polynomials--
26. The diagonal terms--
27. The off-diagonal terms--
28. Conclusion--
29. Computations and the optimal mollifier--
Appendix A. Smooth bump functions--
Appendix B. The gamma function--
Bibliography--
Index.
This 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.
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