A Few Theorems on Primes

Sengupta, Sroyon (2022) A Few Theorems on Primes. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS dissertation of Sroyon Sengupta (17MS076)) 17MS076_Thesis_file.pdf - Submitted Version Restricted to Repository staff only Download (716kB)

Abstract

This report consists of three separate parts, comprising of four important results on prime numbers that are critical in the study of contemporary number theory. The first part contains the complete analytical proof of the Prime Number Theorem. We study several important properties of the Riemann Zeta function in due course using complex analytic tools to prove the theorem. The second part includes the proof of the Vinogradov’s Three Primes Theorem, which involves the use of Hardy-Littlewood Circle Method. We study major arcs and minor arcs and use them to give the proof. In the last part, we reproduce the paper Primes in Tuples I by D. Goldston, J. Pintz and C. Yildirim, where we discuss on gaps between primes and how close consecutive primes are to each other. We use the GPY Seive method to prove two ground-breaking results given in the paper. Finally, we discuss a few further developments that are possible and have happened in the corresponding topic.

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