Lectures on the Riemann zeta function / H. Iwaniec.

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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...

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Bibliographic Details
Uniform Title:University lecture series (Providence, R.I.) ; 62.
Main Author: Iwaniec, Henryk
Language:English
Published: Providence, Rhode Island : American Mathematical Society, [2014]
Series:University lecture series (Providence, R.I.) ; 62.
Subjects:
Riemann, Bernhard, > 1826-1866.
Functions, Zeta.
Riemann hypothesis.
Numbers, Prime.
Number theory > Multiplicative number theory > Distribution of primes.
Number theory > Multiplicative number theory > Asymptotic results on arithmetic functions.
Riemannsche Zetafunktion.
Physical Description:vii, 119 pages : illustrations ; 26 cm.
Variant Title:
Riemann zeta function
Format: Book
Contents:
  • 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
  • 15. Zeros on 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.