No description available.

In this thesis, we will study a class of divisor functions: the prime symmetric functions. These are polynomials over Q in the so-called elementary prime symmetric functions, whose values lie in Z. The latter are defined on the nonnegative integers and take the values of the elementary symmetric functions applied to the multi-set of prime factors (with repetition) of an integer n. Initially we look at basic properties of prime symmetric functions, and consider analogues of questions posed for the usual sum of proper divisors function, such as those concerning perfect numbers or Aliquot sequences. We consider the inverse question of when, and in how many ways a number $n$ can be expressed as f(m) for certain prime symmetric functions f. Then we look at asymptotic formulae for the average orders of certain fundamental prime symmetric functions, such as the arithmetic function whose value at n is the sum of k-th powers of the prime divisors (with repetition) of n. For these last functions in particular, we also look at statistical results by comparing their distribution of values with the distribution of the largest prime factor dividing n. In addition to average orders, we look at the modular distribution of prime symmetric functions, and show that for a fundamental class, they are uniformly distributed over any fixed modulus. Then our focus shifts to the related area of partitions into prime powers. We compute the appropriate asymptotic formulae, and demonstrate important monotonicity properties. We conclude by looking at iteration problems for some of the simpler prime symmetric functions. In doing so, we consider the empirical basis for certain conjectures, and are left with many open problems. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis, we define a certain group of order 160, which we call a hyperpentagonal group, and we prove that every totally real D5-extension of the rationals ramified at only one prime is contained in a hyperpentagonal extension of the rationals. This generalizes a result of Doud and Childers (originally conjectured by Wong) that every totally real S3 extension of the rationals ramified at only one prime is contained in an S4 extension.

This paper discusses the concepts in algebraic and analytic number theory used in the proofs of Dirichlet's and Cheboterev's density theorems. It presents special cases of results due to the latter theorem for which greatly simplified proofs exist. / Master of Science

The aim of this paper is to exhibit analogs in Number Theory of certain well known theorems and methods of the Theory of Equations.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

A number of exciting recent developments in the field of sieve theory have been done concerning bounded gaps between prime numbers. One of the main techniques used in these papers is a modified version of Selberg's Sieve from the 1940's. While there are a number of sources that explain the original sieve, most, if not all, are quite inaccessible to those without significant experience in analytic number theory. The goal of this exposition is to change that. The statement and proof of the general form of Selberg's sieve is, by itself, difficult to understand and appreciate. For this reason, the inital exposition herein will be about one particular application: to recover Chebysheff's upper bound on the order of magnitude of the number of primes less than a given number. As Selberg's sieve follows some of the same initial steps as the more elementary sieve of Eratosthenes, this latter sieve will be worked through as well. To help the reader get a better sense of Selberg's sieve, a few particular applications are worked through, including an upper bound on the number of twin primes less than a number. This will then be used to show the convergence of the reciprocals of the twin primes.

This thesis considers the importance of the inferior parietal lobe for calculation and Arabic number comparison. The first experiment demonstrates that repetitive Transcranial Magnetic Stimulation (rTMS) can be used on normal subjects to replicate findings from studies of patients whose ability to calculate after brain injury was impaired. While subjects were solving addition tasks, rTMS was applied over anterior and posterior areas of the inferior parietal lobule and the adjoining intraparietal sulcus (aIPL+S, pIPL+S). In line with results from patient studies, magnetic stimulation showed a disruptive effect only over left IPL+S. It had no disruptive effect when delivered over right inferior parietal lobule and the adjoining intraparietal sulcus. To investigate the representation of number magnitude in the human brain rTMS was subsequently applied to the same inferior parietal regions while subjects performed a number comparison task. With numbers between 31 and 99, repetitive TMS over the pIPL+S disrupted organisation of the putative "number line". rTMS had no disruptive effect when delivered over aIPL+S, in either the left or right hemisphere. With numbers between 1 and 9, however, TMS over the pIPL+S did not impair task performance. Here, TMS had a disruptive effect when delivered over aIPL+S, in either the left or right hemisphere, thus suggesting that areas in the inferior parietal lobes might be specialised for certain number sizes. The idea of a spatial mental number line was further investigated in a detailed single-case description of a person with an automatic mental number line. In the last experiment, functional Magnetic Resonance Imaging (fMRI) was used to investigate number comparison. The fMRI study gave some indication that small numbers might be represented in the aIPL+S region. In general, the fMRI results suggest that parietal cortical contribution to number magnitude representation is intimately related to its role in basic sensorimotor processes.