# Beurling Generalized Numbers

### About this Title

**Harold G. Diamond**, *University of Illinois, Urbana, IL* and **Wen-Bin Zhang (Cheung Man Ping)**, *University of the West Indies, Kingston, Jamaica*

Publication: Mathematical Surveys and Monographs

Publication Year:
2016; Volume 213

ISBNs: 978-1-4704-3045-0 (print); 978-1-4704-3538-7 (online)

DOI: https://doi.org/10.1090/surv/213

MathSciNet review: MR3559358

MSC: Primary 11N80

### Table of Contents

**Front/Back Matter**

**Chapters**

- Chapter 1. Overview
- Chapter 2. Analytic machinery
- Chapter 3. $\mathbf {dN}$ as an exponential and Chebyshev’s identity
- Chapter 4. Upper and lower estimates of $\mathbf {N(x)}$
- Chapter 5. Mertens’ formulas and logarithmic density
- Chapter 6. O-density of g-integers
- Chapter 7. Density of g-integers
- Chapter 8. Simple estimates of $\mathbf {\pi (x)}$
- Chapter 9. Chebyshev bounds—Elementary theory
- Chapter 10. Wiener-Ikehara Tauberian theorems
- Chapter 11. Chebyshev bounds—Analytic methods
- Chapter 12. Optimality of a Chebyshev bound
- Chapter 13. Beurling’s PNT
- Chapter 14. Equivalences to the PNT
- Chapter 15. Kahane’s PNT
- Chapter 16. PNT with remainder
- Chapter 17. Optimality of the dlVP remainder term
- Chapter 18. The Dickman and Buchstab functions