An annotated bibliography for comparative prime number theory

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained. We encourage feedback on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey" (arXiv:1202.3408

Some explicit results in analytic number theory

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