Application: an Asymptotic Criterion for the Riemann Hypothesis

  • André VorosEmail author
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 8)


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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.CEA-Saclay Institut de Physique Théorique (IPhT) Orme des MerisiersGif-sur-YvetteFrance

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