Application: an Asymptotic Criterion for the Riemann Hypothesis

Abstract

This chapter is the only one whose contents expressly link to the truth or falsity of the Riemann Hypothesis [92]: it deals with a new and sharp criterion for this conjecture. Many statements equivalent to the Riemann Hypothesis already exist; they typically involve inequalities such as infinitely many positivity conditions to be all obeyed without exception; our criterion may have an unusual form in that it gives a neat asymptotic alternative instead (actually in two variants). It links to an earlier criterion by Li, which states that the Riemann Hypothesis is true if and only if a specific real sequence has all its terms positive. We actually use the same sequence, but through a different filter: from the general framework of the previous chapters, we deduce that the Riemann Hypothesis can be expressed purely in terms of the large-n behavior of that sequence.