Some explicit results in analytic number theory

UNSW, Sydney 2023

In this thesis, we present a few explicit results in the field of analytic number theory. The first set of results are in the area of multiplicative number theory which deals with the behaviour of multiplicative arithmetic functions. The second set of results is in the area of additive number theory which is concerned with the additive representation of numbers. Chapter 1 (Introduction) contextualises these results. In Chapter 2 (Results in multiplicative number theory), we look at two main problems. The first problem (Section 2.1) deals with Mertens’ product formula for number fields and generalises results related to oscillations (Diamond & Pintz, 2009) and bias (Lamzouri, 2016) in Mertens’ product formula in the classical setting. The second problem (Section 2.2) is concerned with improving the upper bound to Mertens’ conjecture, building upon the work of Odlyzko, te Riele, Pintz, Kotnik and Saouter. In Chapter 3 (Results in additive number theory), we consider Goldbach-like problems. In Section 3.1, we prove new results pertaining to a previously known approximation to the Goldbach problem (Dudek, 2017) which involves writing an integer as the sum of a prime and a squarefree number. By imposing further divisibility conditions on the squarefree number in this representation, we get a result that is closer to the Goldbach conjecture “in spirit”. In Section 3.2, we prove the best result so far for the Linnik approximation to the problem of representing an even integer as the sum of four squares of primes.

number theory; analytic number theory; goldbach conjecture; mertens conjecture; mertens product formula; circle method; anzsrc-for: 49 MATHEMATICAL SCIENCES; doctoral thesis; http://purl.org/coar/resource_type/c_db06

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UNIV OF NEW S WALES
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