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Journal of the London Mathematical Society Advance Access originally published online on February 13, 2009
Journal of the London Mathematical Society 2009 79(2):422-444; doi:10.1112/jlms/jdn078
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© 2009 London Mathematical Society

Factorial ratios, hypergeometric series, and a family of step functions

Jonathan W. Bober

Department of Mathematics
The University of Michigan
530 Church Street
Ann Arbor, MI 48109
USA

We give a complete classification of a certain family of step functions related to the Nyman–Beurling approach to the Riemann hypothesis and previously studied by Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations have been made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nFn–1. We also note applications to the classification of cyclic quotient singularities.

2000 Mathematics Subject Classification 11B65 (primary), 11M26, 14M25, 33C20 (secondary).

Received September 12, 2007; revised September 2, 2008; published online February 13, 2009.


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