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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn052, 19 pages, doi:10.1093/imrn/rnn052 published on June 11, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Quotient Singularities, Integer Ratios of Factorials, and the Riemann Hypothesis

Alexander Borisov

Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

Correspondence: Correspondence to be sent to: borisov{at}pitt.edu

The goal of this paper is to reveal a close connection between the following three subjects that have not been studied together in the past:

  1. terminal and canonical cyclic quotient singularities;
  2. integer ratios of factorials;
  3. Nyman's approach to the Riemann hypothesis.

In particular, we notice that the list of the 29 stable quintuples of Mori–Morrison–Morrison coincides, up to the choice of notation, with the list of the 29 step-functions with five terms of Vasyunin. By the work of Rodriguez Villegas and Bober, they are also connected to the algebraic hypergeometric functions. These unexpected connections lead to several interesting open questions.


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