Probabilistic Number Theory II: Central Limit Theorems

Description

In this volume we study the value distribution of arithmetic functions, allowing unbounded renormalisations. The methods involve a synthesis of Probability and Number Theory; sums of independent infinitesimal random variables playing an important role. A central problem is to decide when an additive arithmetic function fin) admits a renormalisation by real functions a(x) and {3(x) > 0 so that asx ~ 00 the frequencies vx(n;f (n) - a(x) :s;; z {3 (x) ) converge weakly; (see Notation). In contrast to volume one we allow {3(x) to become unbounded with x. In particular, we investigate to what extent one can simulate the behaviour of additive arithmetic functions by that of sums of suit­ ably defined independent random variables. This fruiful point of view was intro­ duced in a 1939 paper of Erdos and Kac. We obtain their (now classical) result in Chapter 12. Subsequent methods involve both Fourier analysis on the line, and the appli­ cation of Dirichlet series. Many additional topics are considered. We mention only: a problem of Hardy and Ramanujan; local properties of additive arithmetic functions; the rate of convergence of certain arithmetic frequencies to the normal law; the arithmetic simulation of all stable laws. As in Volume I the historical background of various results is discussed, forming an integral part of the text. In Chapters 12 and 19 these considerations are quite extensive, and an author often speaks for himself. Front Matter....Pages i-xviii Unbounded Renormalisations: Preliminary Results....Pages 1-11 The Erdös-Kac Theorem. Kubilius Models....Pages 12-51 The Weak Law of Large Numbers. I....Pages 52-57 The Weak Law of Large Numbers. II....Pages 58-97 A Problem of Hardy and Ramanujan....Pages 98-121 General Laws for Additive Functions. I: Including the Stable Laws....Pages 122-146 The Limit Laws and the Renormalising Functions....Pages 147-183 General Laws for Additive Functions. II : Logarithmic Renormalisation....Pages 184-210 Quantitative Mean-Value Theorems....Pages 211-261 Rate of Convergence to the Normal Law....Pages 262-289 Local Theorems for Additive Functions....Pages 290-312 The Distribution of the Quadratic Class Number....Pages 313-329 Problems....Pages 330-341 Back Matter....Pages I-XXXVI