Number theory II.a - The circle method

This course provides an introduction to the Hardy-Littlewood circle method. After reviewing some fundamental concepts from Fourier analysis, our goal is to demonstrate the power of the circle method in addressing additive equations and additive patterns.

Topics covered include

• Fourier inversion formula and Parseval's identity, which serve as the foundational tools of the course
• Waring's problem, which shows that every sufficiently large natural number (under suitable local conditions) can be expressed as the sum of a fixed number of k-th powers of natural numbers.
• Vinogradov's three primes theorem, which proves that every sufficiently large odd number can be written as the sum of three prime numbers.
• Roth's theorem, which states that any subset of the natural numbers with positive density contains a non-trivial three-term arithmetic progression.
• Roth's theorem in the primes, which extends the previous result to subsets of the primes with positive relative density.