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Probabilistic Number Theory II

Central Limit Theorems

  • P. D. T. A. Elliott

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

Table of contents

  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 147-183
  8. P. D. T. A. Elliott
    Pages 211-261
  9. P. D. T. A. Elliott
    Pages 262-289
  10. P. D. T. A. Elliott
    Pages 290-312
  11. P. D. T. A. Elliott
    Pages 313-329
  12. P. D. T. A. Elliott
    Pages 330-341
  13. Back Matter
    Pages I-XXXVI

About this book

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 Springer Book Archive
  • Print ISBN 978-1-4612-9994-3
  • Online ISBN 978-1-4612-9992-9
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site
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