Arithmetic Functions and Integer Products

  • P. D. T. A. Elliott

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 272)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Introduction

    1. P. D. T. A. Elliott
      Pages 1-17
  3. First Motive

    1. Front Matter
      Pages 19-21
    2. P. D. T. A. Elliott
      Pages 23-36
    3. P. D. T. A. Elliott
      Pages 37-52
    4. P. D. T. A. Elliott
      Pages 53-77
    5. P. D. T. A. Elliott
      Pages 78-80
    6. P. D. T. A. Elliott
      Pages 81-95
  4. Second Motive

    1. Front Matter
      Pages 97-100
    2. P. D. T. A. Elliott
      Pages 101-120
    3. P. D. T. A. Elliott
      Pages 155-175
  5. Third Motive

    1. Front Matter
      Pages 177-181
    2. P. D. T. A. Elliott
      Pages 183-203
    3. P. D. T. A. Elliott
      Pages 204-243
    4. P. D. T. A. Elliott
      Pages 244-249
    5. P. D. T. A. Elliott
      Pages 250-258
    6. P. D. T. A. Elliott
      Pages 259-263
    7. P. D. T. A. Elliott
      Pages 264-276
    8. P. D. T. A. Elliott
      Pages 277-290

About this book

Introduction

Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ± I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs. Assuming hex) to be non­ negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -x». Except for a renormalization this is the well-known function of Shannon. What do these results have in common? They both apply the theory of arithmetic functions. The two widest classes of arithmetic functions are the real-valued additive and the complex-valued multiplicative functions. Beginning in the thirties of this century, the work of Erdos, Kac, Kubilius, Turan and others gave a discipline to the study of the general value distribution of arithmetic func­ tions by the introduction of ideas, methods and results from the theory of Probability. I gave an account of the resulting extensive and still developing branch of Number Theory in volumes 239/240 of this series, under the title Probabilistic Number Theory.

Keywords

Arithmetic Functions Lemma Prime Prime number algebra number theory

Authors and affiliations

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

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4613-8548-6
  • Copyright Information Springer-Verlag New York 1985
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-8550-9
  • Online ISBN 978-1-4613-8548-6
  • Series Print ISSN 0072-7830
  • About this book