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.

Zusammenfassung
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.

Inhalt
Duality 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.