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Grundlehren der mathematischen Wissenschaften

Arithmetic Functions and Integer Products

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

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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.

Table of contents (25 chapters)

Buy this book

eBook $59.99
price for USA in USD (gross)
  • ISBN 978-1-4613-8548-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $99.00
price for USA in USD
  • ISBN 978-0-387-96094-4
  • Free shipping for individuals worldwide
  • Currently out of stock.
Softcover $79.99
price for USA in USD
  • ISBN 978-1-4613-8550-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Arithmetic Functions and Integer Products
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
272
Copyright
1985
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York Inc.
eBook ISBN
978-1-4613-8548-6
DOI
10.1007/978-1-4613-8548-6
Hardcover ISBN
978-0-387-96094-4
Softcover ISBN
978-1-4613-8550-9
Series ISSN
0072-7830
Edition Number
1
Number of Pages
461
Topics