# Probabilistic Number Theory II

## Central Limit Theorems

• P. D. T. A. Elliott
Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 240)

1. Front Matter
Pages i-xviii
2. P. D. T. A. Elliott
Pages 1-11
3. P. D. T. A. Elliott
Pages 12-51
4. P. D. T. A. Elliott
Pages 52-57
5. P. D. T. A. Elliott
Pages 58-97
6. P. D. T. A. Elliott
Pages 98-121
7. P. D. T. A. Elliott
Pages 122-146
8. P. D. T. A. Elliott
Pages 147-183
9. P. D. T. A. Elliott
Pages 184-210
10. P. D. T. A. Elliott
Pages 211-261
11. P. D. T. A. Elliott
Pages 262-289
12. P. D. T. A. Elliott
Pages 290-312
13. P. D. T. A. Elliott
Pages 313-329
14. P. D. T. A. Elliott
Pages 330-341
15. Back Matter
Pages I-XXXVI

### Introduction

In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suit­ ably defined independent random variables. This fruiful point of view was intro­ duced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the appli­ cation of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself.

### Keywords

Prime Prime number Wahrscheinlichkeitstheoretische Zahlentheorie calculus number theory

#### Authors and affiliations

• P. D. T. A. Elliott
• 1
1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-9992-9
• Copyright Information Springer-Verlag New York 1980
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4612-9994-3
• Online ISBN 978-1-4612-9992-9
• Series Print ISSN 0072-7830