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Author Elliott, P. D. T. A. author Arithmetic Functions and Integer Products [electronic resource] / by P. D. T. A. Elliott New York, NY : Springer New York, 1985 http://dx.doi.org/10.1007/978-1-4613-8548-6 461p. online resource

SUMMARY

Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = ยฑ 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 nonยญ negative, and to satisfy certain standard properties, it must have the form -A(x log x + (I - x) 10g(I -xยป. 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 funcยญ 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

CONTENT

Duality and the Differences of Additive Functions -- First Motive -- 1 Variants of Well-Known Arithmetic Inequalities -- 2 A Diophantine Equation -- 3 A First Upper Bound -- 4 Intermezzo: The Group Q*/? -- 5 Some Duality -- Second Motive -- 6 Lemmas Involving Prime Numbers -- 7 Additive Functions on Arithmetic Progressions with Large Moduli -- 8 The Loop -- Third Motive -- 9 The Approximate Functional Equation -- 10 Additive Arithmetic Functions on Differences -- 11 Some Historical Remarks -- 12 From L2 to L? -- 13 A Problem of Kรกtai -- 14 Inequalities in L? -- 15 Integers as Products -- 16 The Second Intermezzo -- 17 Product Representations by Values of Rational Functions -- 18 Simultaneous Product Representations by Values of Rational Functions -- 19 Simultaneous Product Representations with aix + bi -- 20 Information and Arithmetic -- 21 Central Limit Theorem for Differences -- 22 Density Theorems -- 23 Problems -- Supplement Progress in Probabilistic Number Theory -- References

Mathematics Number theory Mathematics Number Theory

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