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Lectures on the Riemann Zeta Function
About this Title
H. Iwaniec, Rutgers University, Piscataway, NJ
Publication: University Lecture Series
Publication Year:
2014; Volume 62
ISBNs: 978-1-4704-1851-9 (print); 978-1-4704-1891-5 (online)
DOI: https://doi.org/10.1090/ulect/062
MathSciNet review: MR3241276
MSC: Primary 11N05; Secondary 11N37
ReadershipTable of Contents
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Front/Back Matter
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
- F. Carlson, Über die Nullstellen der Dirichletschen Reihen und der Riemannschen $\zeta$-Funktion., Ark. Mat. Astron. Fys. 15 (1921), no. 20, 28.
- J. B. Conrey, On the distribution of the zeros of the Riemann zeta-function, Topics in Analytic Number Theory (Austin, Tex., 1982), Univ. Texas Press, Austin, TX, 1985, pp. 28–41.
- —, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J. Reine Angew. Math. 399 (1989), 1–26.
- D. W. Farmer, Long mollifiers of the Riemann zeta-function, Mathematika 40 (1993), no. 1, 71–87.
- S. Feng, Zeros of the Riemann zeta function on the critical line, J. Number Theory 132 (2012), no. 4, 511–542.
- D. R. Heath-Brown, Simple zeros of the Riemann zeta-function on the critical line, Bull. London Math. Soc. 11 (1979), 17–18.
- G. H. Hardy and J. E. Littlewood, The zeros of Riemann’s zeta-function on the critical line, Math. Z. 10 (1921), no. 3-4, 283–317.
- N. Levinson, More than one third of zeros of Riemann’s zeta-function are on $\sigma =1/2$, Advances in Math. 13 (1974), 383–436.
- B. Riemann, Ãber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Akad. Berlin (1859), 671–680.
- A. Selberg, On the zeros of Riemann’s zeta-function on the critical line, Arch. Math. Naturvid. 45 (1942), no. 9, 101–114.