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Type: Thesis
Title: Distribution of additive functions in algebraic number fields
Author: Hughes, Garry
Issue Date: 1987
School/Discipline: Dept. of Pure Mathematics
Abstract: 1. Background. This thesis is based on ideas drawn from classical probabilistic number theory, from the work of Novoselov [2], and from the relevant work on algebraic number fields. Classical probabilistic number theory (as described in Elliott [1], for example) is concerned with the distribution of arithmetic functions on the ring of (rational) integers, ZZ. Two well-known results in this area are the Hardy-Ramanujan and the Erdos-Wintner theorems. The Hardy-Ramanujan theorem states that, in some sense, every integer n has about log log n prime divisors, and the Erdos-Wintner theorem gives conditions under·which additive functions have limiting distributions. The original proofs of both results were subsequently considerably simplified by using a result known as the Tunin-Kubilius inequality. Although results in this field have a definite probabilistic flavour, it has not proved easy to establish them by a direct appeal to the theory of probability. Novoselov [2] developed a probability space which provides a natural framework for developing results of probabilistic number theory from results of probability. For example, using standard results from probability theory and some arithmetic estimates (which amount to the Turan-Kubilius inequality) he obtained the Hardy-Ramanujan and Erdos-Wintner theorems. Many of the results of probabilistic number theory have been generalized to results concerning the distribution of additive functions on the ideals of the ring, V, of integers of an algebraic number field (see Prachar [3], for example). However, work in this area has not used a probabilistic framework as fully as in the classical case of ZZ. 2. Aims. The aim of this thesis is to set up a space for probabilistic number theory in algebraic number fields analogous to that of Novoselov [2] for ZZ and to apply his approach to develop analogues in 1) of the Hardy-Ramanujan and Erdos-Wintner theorems. We endeavour to produce as much as possible without the use of sieve results. 3. Contents Chapter 1 of this thesis is an introduction to the background outlined above, and Chapter 2 gathers together some preliminary material. In Chapter 3 we obtain an analogue of the Tunin-Kubilius inequality in D. For this purpose we estimate the number of elements of an ideal which lie in a multiple of the fundamental domain of D (viewed as a lattice). In Chapter 4 we construct a probability space Ω, containing D, using two different approaches. One approach is analogous to that in Novoselov [2). The other views n as the product of the completions of D with respect to its non-Archimedean valuations and this enables us to simplify some proofs. rn· Chapter 5 we prove versions of the Hardy-Ramanujan and ErdosWintner theorems for additive functions on the principal ideals of D. Some examples are discussed. In Chapter 6 we consider additive functions on all the ideals of 1) (not just the principal ideals). We prove Prachar's version of the Hardy-Ramanujan theorem (see Prachar [3]) by using the results of Chapter 5 and the correspondence between the ideals of D in a given class and certain elements of V. References [1] Elliott,P.D.T.A., Probabilistic number theory, Volumes I and 11. Springer-Verlag, New York (1979). [2] Novoselov,E.V., A new method in probabilistic number theory. Amer. Math. Soc. Translations (2) 52 (1966), pp. 217-275. (3] Prachar,K., Verallgemeinerung eines Satzes 'IJOn Hardy und Ramanujan auf algebraische Zahlkorper. Monatsh. Math. 56 (1952), pp. 229-232.
Advisor: Pitman, Jane
Dissertation Note: Thesis (MSc) -- University of Adelaide, Dept. of Pure Mathematics, 1987
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