Arithmetic Functions and Integer Products

ELLIOTT, P. D. T. A.

ISBN 10: 0387960945 / ISBN 13: 9780387960944

Editore: Springer, New York, 1984

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Die Grundlehren der mathematischen Wissenschaften/A Series of Comprehensive Studies in Mathematics 272. xv, 461 p. 24 cm. Yellow cloth. Ex library with labels on spine and rear pastedown, ink stamps on top edge and title. Spine and parts of boards faded. Dents in lower edges. Codice articolo 147383

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Titolo: Arithmetic Functions and Integer Products

Casa editrice: Springer, New York

Data di pubblicazione: 1984

Legatura: Hardcover

Condizione: ex library-good

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Riassunto:

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.