Pretentious approach to analytic number theory

• Thesis title in Czech: Předstírající přístup k analytické teorii čísel
• Thesis title in English: Pretentious approach to analytic number theory
• Key words: analytická teorie čísel, předstírající přístup, aritmetická funkce, rozložení prvočísel
• English key words: analytic number theory, pretentious approach, arithmetic function, distribution of prime numbers
• Academic year of topic announcement: 2017/2018
• Thesis type: diploma thesis
• Thesis language: angličtina
• Department: Department of Algebra (32-KA)
• Supervisor: doc. Mgr. Vítězslav Kala, Ph.D.
• Author: hidden - assigned and confirmed by the Study Dept.
• Date of registration: 01.10.2017
• Date of assignment: 01.10.2017
• Confirmed by Study dept. on: 27.11.2017
• Date and time of defence: 14.06.2018 09:00
• Date of electronic submission: 09.05.2018
• Date of submission of printed version: 11.05.2018
• Date of proceeded defence: 14.06.2018
• Opponents: RNDr. Jaroslav Hančl, Ph.D.

Guidelines

The recently developed "pretentious approach" to analytic number theory studies the asymptotic behavior of arithmetic functions using Halasz's theorem from 1975. This allows one to prove a number of important results in analytic number theory without relying on the analytic continuation of the Riemann zeta function. The goal of the thesis is to study and compare the classical and pretentious proofs of such theorems, such as the prime number theorem, Siegel-Walfisz theorem or Halasz theorem, and possibly try to further generalize some of these results.

References

A. Granville, A. J. Harper, K. Soundararajan: A more intuitive proof of a sharp version of Halász's theorem, 2017, arXiv:1706.03755

D. Koukoulopoulos: Pretentious multiplicative functions and the prime number theorem for arithmetic progressions, Compos. Math., 149 (2013), no. 7, 1129-1149

D. Koukoulopoulos: The distribution of prime numbers, online lecture notes, http://dms.umontreal.ca/~koukoulo/documents/notes/primes.pdf