The Least Quadratic Nonresidue and Vinogradov's Hypothesis

Abstract

These are rough notes covering the second block of lectures in the “Elementary Methods in Analytic Number Theory” course. In these lectures we will develop several forms of the large sieve inequality, which assert that no sequence can be well correlated with many exponentials or poorly distributed in many arithmetic progressions. By combining the large sieve with Vaughan’s Identity and the Siegel– Walfisz theorem, we will deduce the Bombieri–Vinogradov theorem on the average distribution of primes in progressions. (No originality is claimed for any of the contents of these notes. In particular, they borrow from the books of Davenport [1] and Iwaniec and Kowalski