# Arithmetic Functions and Integer Products

• P. D. T. A. Elliott
Book

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

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
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Pages 53-77
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Pages 78-80
6. P. D. T. A. Elliott
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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 121-154
4. 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
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4. P. D. T. A. Elliott
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8. P. D. T. A. Elliott
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### 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
• Print ISBN 978-1-4613-8550-9
• Online ISBN 978-1-4613-8548-6
• Series Print ISSN 0072-7830