Alfréd Rényi, the Density of L-Zeros and the Goldbach Conjecture

Alfréd Rényi, the founding director of the Mathematical Institute of the Hungarian Academy of Sciences was the first mathematician who proved a density theorem for the zeros of Dirichlet’s 𝐿-functions with variable moduli. This was based on a refinement of the large sieve of Linnik, developed by Rényi himself. He used this to show a weaker form of the binary Goldbach conjecture. His density theorem was the first forerunner of the famous Bombieri–Vinogradov theorem. We give a simple alternative proof of a weaker form of the Bombieri–Vinogradov theorem, based only on classical facts about 𝐿-functions (including Siegel’s theorem) and a simple but ingenious idea of Halász, but without using any form of the large sieve.