The Siegel zero effect

Abstract

When dealing with the Brun-Titchmarsh Theorem (Theorem 2.2 of this monograph), and in general, with sieve methods, the question of the connections between the parity principle, the constant 2 in this theorem and the so-called Siegel zeros cannot be avoided. (Selberg, 1949) shows that the constant 2 + o(1) in the Brun-Titchmarsh Theorem is optimal, if we stick to a sieve method in a fairly general context. He expanded this theory into what is known as the “parity principle” in (Selberg, 1972). See also (Bombieri, 1976). However, this objection is methological and belongs much more to the realm of the combinatorial sieve. In the restricted framework of the Brun-Tichmarsh Theorem, or in the even more restricted framework of this Theorem for the initial interval only, the constant 2 and “the parity principle” are indeed two different issues. This chapter is first devoted to links and parallels between Siegel zeros and the constant 2 in the aforementioned Theorem.