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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 ContentsDuality and the Differences of Additive Functions.- First Motive.- 1 Variants of Well-Known Arithmetic Inequalities.- Multiplicative Functions.- Generalized Turán-Kubilius Inequalities.- Selberg’s Sieve Method.- Kloosterman Sums.- 2 A Diophantine Equation.- 3 A First Upper Bound.- The First Inductive Proof.- The Second Inductive Proof.- Concluding Remarks.- 4 Intermezzo: The Group Q*/?.- 5 Some Duality.- Duality in Finite Spaces.- Self-adjoint Maps.- Duality in Hilbert Space.- Duality in General.- Second Motive.- 6 Lemmas Involving Prime Numbers.- The Large Sieve and Prime Number Sums.- The Method of Vinogradov in Vaughan’s Form.- Dirichlet L-Series.- 7 Additive Functions on Arithmetic Progressions with Large Moduli.- Additive Functions on Arithmetic Progressions.- Algebraicanalytic Inequalities.- 8 The Loop.- Third Motive.- 9 The Approximate Functional Equation.- 10 Additive Arithmetic Functions on Differences.- The Basic Inequality.- The Decomposition of the Mean.- Concluding Remarks.- 11 Some Historical Remarks.- 12 From L2 to L?.- 13 A Problem of Kátai.- 14 Inequalities in L?.- 15 Integers as Products.- More Duality; Additive Functions as Characters.- Divisible Groups and Modules.- Sets of Uniqueness.- Algorithms.- 16 The Second Intermezzo.- 17 Product Representations by Values of Rational Functions.- A Ring of Operators.- Practical Measures.- 18 Simultaneous Product Representations by Values of Rational Functions.- Linear Recurrences in Modules.- Elliptic Power Sums.- Concluding Remarks.- 19 Simultaneous Product Representations with aix + bi.- 20 Information and Arithmetic.- Transition to Arithmetic.- Information as an Algebraic Object.- 21 Central Limit Theorem for Differences.- 22 Density Theorems.- Groups of Bounded Order.- Measures on Dual Groups.- Arithmic Groups.- Concluding Remarks.- 23 Problems.- Exercises.- Unsolved Problems.- Supplement Progress in Probabilistic Number Theory.- Analogues of the Turán-Kubilius Inequality.- References.