Arithmetic Functions and Integer Products Hardback
by P.D.T.A. Elliott
Part of the Grundlehren der mathematischen Wissenschaften series
Hardback
Description
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.
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Item not Available
- Format:Hardback
- Pages:461 pages, 461 p.
- Publisher:Springer-Verlag New York Inc.
- Publication Date:20/11/1984
- Category:
- ISBN:9780387960944
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Information
-
Item not Available
- Format:Hardback
- Pages:461 pages, 461 p.
- Publisher:Springer-Verlag New York Inc.
- Publication Date:20/11/1984
- Category:
- ISBN:9780387960944
