Probabilistic Number Theory II

1. Front Matter

   Pages i-xviii

   PDF

2. Unbounded Renormalisations: Preliminary Results

   • P. D. T. A. Elliott

   Pages 1-11

3. The Erdös-Kac Theorem. Kubilius Models

   • P. D. T. A. Elliott

   Pages 12-51

4. The Weak Law of Large Numbers. I

   • P. D. T. A. Elliott

   Pages 52-57

5. The Weak Law of Large Numbers. II

   • P. D. T. A. Elliott

   Pages 58-97

6. A Problem of Hardy and Ramanujan

   • P. D. T. A. Elliott

   Pages 98-121

7. General Laws for Additive Functions. I: Including the Stable Laws

   • P. D. T. A. Elliott

   Pages 122-146

8. The Limit Laws and the Renormalising Functions

   • P. D. T. A. Elliott

   Pages 147-183

9. General Laws for Additive Functions. II : Logarithmic Renormalisation

   • P. D. T. A. Elliott

   Pages 184-210

10. Quantitative Mean-Value Theorems

    • P. D. T. A. Elliott

    Pages 211-261

11. Rate of Convergence to the Normal Law

    • P. D. T. A. Elliott

    Pages 262-289

12. Local Theorems for Additive Functions

    • P. D. T. A. Elliott

    Pages 290-312

13. The Distribution of the Quadratic Class Number

    • P. D. T. A. Elliott

    Pages 313-329

14. Problems

    • P. D. T. A. Elliott

    Pages 330-341

15. Back Matter

    Pages I-XXXVI

    PDF

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.

• Prime
• Prime number
• Wahrscheinlichkeitstheoretische Zahlentheorie
• calculus
• number theory

• Department of Mathematics, University of Colorado, Boulder, USA

  P. D. T. A. Elliott

Policies and ethics