Prime Gaps and Multiplicative Sieve Inequalities

This lecture discusses the theorem of Bombieri-Vinogradov, focusing on the error term for the prime counting function in arithmetic progressions. The instructor outlines the process of reducing the proof of this theorem by expressing the error term in terms of sums over multiplicative characters. The discussion includes the significance of primitive characters and the conditions under which the error term remains small. The lecture also introduces the multiplicative large sieve inequality, explaining its application in bounding sums over large moduli. The instructor emphasizes the importance of controlling the localization of zeros in Dirichlet characters and how this relates to the overall proof structure. Additionally, the lecture covers corollaries derived from the large sieve inequality, illustrating how these results can be utilized to achieve bounds on sums involving prime gaps. The session concludes with a discussion on the implications of these results for understanding the distribution of prime numbers and the multiplicative structure of arithmetic functions.