Investigation of the Circle Method: its origin and some applications

Drummond, Errol (2020) Investigation of the Circle Method: its origin and some applications. Bachelor's Thesis, Mathematics.

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Abstract

The circle method was developed by Hardy & Littlewood in their attempts to apply analytic function theory to Waring's problem, which asks for the smallest integer s such that all sufficiently large integers can be written as a sum of s k'th powers. The method can be applied to problems of an additive nature, and in particular the method was used to prove Goldbach's weak conjecture for sufficiently large integers. This was first done by Hardy & Littlewood, but they assumed the generalised Riemann hypothesis; Vinogradov managed to use the method without this assumption to prove the same result. This thesis, after introducing the main ideas behind the method and how it works, explores how it can be applied to prove Vinogradov's theorem, that the weak Goldbach conjecture is true for all sufficiently large integers. Following this, there is a short look at how the method can be applied to proving that all positive integers can be written as a sum of three triangular numbers.

Item Type:	Thesis (Bachelor's Thesis)
Supervisor name:	Top, J. and Gajovic, S.
Degree programme:	Mathematics
Thesis type:	Bachelor's Thesis
Language:	English
Date Deposited:	13 Jul 2020 13:18
Last Modified:	13 Jul 2020 13:18
URI:	https://fse.studenttheses.ub.rug.nl/id/eprint/22624

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