Some topics in analytic and probabilistic number theory

This thesis is not available on this repository until the author agrees to make it public. If you are the author of this thesis and would like to make your work openly available, please contact us: thesis@repository.cam.ac.uk.The Library can supply a digital copy for private research purposes; interested parties should submit the request form here: http://www.lib.cam.ac.uk/collections/departments/digital-content-unit/ordering-imagesPlease note that print copies of theses may be available for consultation in the Cambridge University Library's Manuscript reading room. Admission details are at http://www.lib.cam.ac.uk/collections/departments/manuscripts-university-archivesThis dissertation studies four problems in analytic and probabilistic number theory. Two of the problems are about a certain random number theoretic object, namely a random multiplicative function. The other two problems are about smooth numbers (i.e. numbers only having small prime factors), both in their own right and in their application to finding solutions to S-unit equations over the integers. Thus all four problems are concerned, in different ways, with _understanding the multiplicative structure of the integers. More precisely, we will establish that certain sums of a random multiplicative function satisfy a normal approximation (i.e. a central limit theorem) , but that the complete sum over all integers less than x does not satisfy such an approximation. This reflects certain facts about the number and size of the prime factors of a typical integer. Our proofs use martingale methods, as well as a conditioning argument special to this problem. Next, we will prove an almost sure omega result for the sum of a random multiplicative function, substantially improving the existing result of Halasz. We will do this using a connection between sums of a random multiplicative function and a certain random trigonometric sum process, so that the heart of our work is proving precise results about the suprema of a class of Gaussian random processes. Switching to the study of smooth numbers, we will establish an equidistribution result for the y-smooth numbers less than x among arithmetic progressions to modulus q, asymptotically as (logx)/(logq)-+ oo, subject to a certain condition on the relative sizes of y and q. The main point of this work is that it does not require any restrictions on the relative sizes of x and y. Our proofs use a simple majorant principle for trigonometric sums, together with general tools such as a smoothed explicit formula. Finally, we will prove lower bounds for the possible number of solutions of some S-unit equations over the integers. For example, we will show that there exist arbitrarily large sets S of prime numbers such that the equation a+ l = c has at least exp{(#S)116- �} solutions (a, c) with all their prime factors from S. We will do this by using discrete forms of the circle method, and the multiplicative large sieve, to count the solutions of certain auxiliary linear equations