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Arithmetic Functions and Integer Products

  • Book
  • © 1985

Overview

Authors:
  1. P. D. T. A. Elliott

    1. Department of Mathematics, University of Colorado, Boulder, USA

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

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Table of contents (25 chapters)

  1. Front Matter

    Pages i-xv
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  2. Introduction

    1. Introduction

      • P. D. T. A. Elliott
      Pages 1-17

  3. First Motive

    1. Front Matter

      Pages 19-21
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    2. Variants of Well-Known Arithmetic Inequalities

      • P. D. T. A. Elliott
      Pages 23-36

    3. A Diophantine Equation

      • P. D. T. A. Elliott
      Pages 37-52

    4. A First Upper Bound

      • P. D. T. A. Elliott
      Pages 53-77

    5. Intermezzo: The Group Q*/Γ

      • P. D. T. A. Elliott
      Pages 78-80

    6. Some Duality

      • P. D. T. A. Elliott
      Pages 81-95

  4. Second Motive

    1. Front Matter

      Pages 97-100
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    2. Lemmas Involving Prime Numbers

      • P. D. T. A. Elliott
      Pages 101-120

    3. Additive Functions on Arithmetic Progressions with Large Moduli

      • P. D. T. A. Elliott
      Pages 121-154

    4. The Loop

      • P. D. T. A. Elliott
      Pages 155-175

  5. Third Motive

    1. Front Matter

      Pages 177-181
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    2. The Approximate Functional Equation

      • P. D. T. A. Elliott
      Pages 183-203

    3. Additive Arithmetic Functions on Differences

      • P. D. T. A. Elliott
      Pages 204-243

    4. Some Historical Remarks

      • P. D. T. A. Elliott
      Pages 244-249

    5. From L 2 to L ∞

      • P. D. T. A. Elliott
      Pages 250-258

    6. A Problem of Kátai

      • P. D. T. A. Elliott
      Pages 259-263

    7. Inequalities in L∞

      • P. D. T. A. Elliott
      Pages 264-276

    8. Integers as Products

      • P. D. T. A. Elliott
      Pages 277-290
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Keywords

About this book

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.

Authors and Affiliations

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

    P. D. T. A. Elliott

Bibliographic Information

  • Book Title: Arithmetic Functions and Integer Products

  • Authors: P. D. T. A. Elliott

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-1-4613-8548-6

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1985

  • Hardcover ISBN: 978-0-387-96094-4Due: 20 November 1984

  • Softcover ISBN: 978-1-4613-8550-9Published: 18 October 2011

  • eBook ISBN: 978-1-4613-8548-6Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: 461

  • Topics: Number Theory

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