Quotient Singularities, Integer Ratios of Factorials, and the Riemann Hypothesis
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:
- terminal and canonical cyclic quotient singularities;
- integer ratios of factorials;
- 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.
References
- Báez-Duarte L. A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis. Matematica e Applicazioni (2003) 14(no. 1):5–11.
- Bell J., Bober J. A note on integer factorial ratios and certain step functions. International Journal of Number Theory. forthcoming.
- Beukers F., Heckman G. Monodromy for the hypergeometric function nFn–1. Inventiones Mathematicae (1989) 95(no. 2):325–54.[CrossRef][ISI]
- Bober J. Factorial ratios, hypergeometric series, and a family of step functions. Submitted to the London Mathematical Society.
- Bombieri E. The Riemann hypothesis. The Millennium Prize Problems (2006) Cambridge, MA: Clay Mathematics Institute. 107–24.
- Bombieri E., Bourgain J. On a conjecture of Borisov. (2007) preprint.
- Borisov A. Convex lattice polytopes and cones with few lattice points inside, from a birational geometry viewpoint. Preprint http://front.math.ucdavis.edu/math.AG/0001109.
- Borisov A. Minimal discrepancies of toric singularities. Manuscripta Mathematica (1997) 92(no. 1):33–45.[CrossRef][ISI]
- Borisov A. On classification of toric singularities, Algebraic geometry, 9. Journal of Mathematical Sciences (New York) (1999) 94(no. 1):1111–3.[CrossRef]
- Clemens H., Kollár J., Mori S. Higher-dimensional complex geometry. Astérisque (1988) 166.
- Gessel I. M., Xin G. A combinatorial interpretation of the numbers 6(2n)!/n!(n+2)! Journal of Integer Sequences (2005) 8(no. 2):1–13.
- Ishida, Masa-Nori, Iwashita N. Canonical cyclic quotient singularities of dimension three. Complex Analytic Singularities. 135–51. Advanced Studies in Pure Mathematics 8. New York: North-Holland, 1986.
- Kollár J., et al. Flips and abundance for algebraic threefolds. Astérisque (1992) 211.
- Landau E. Sur les conditions de divisibilité d'un produit de factorielles par un autre. Collected Works. 1–116. Vol. 1. Essen, Germany: Thales, 1985.
- Lawrence J. Finite unions of closed subgroups of the n-dimensional torus. Applied Geometry and Discrete Mathematics. 433–41. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 4. Providence, RI: American Mathematical Society, 1991.
- Mori S., Morrison D. R., Morrison I. On four-dimensional terminal quotient singularities. Mathematics of Computation (1988) 51(no. 184):769–86.[CrossRef][ISI]
- Morrison D. R. Canonical quotient singularities in dimension three. Proceedings of the American Mathematical Society (1985) 93(no. 3):393–6.[CrossRef][ISI]
- Morrison D. R., Stevens G. Terminal quotient singularities in dimensions three and four. Proceedings of the American Mathematical Society (1984) 90(no. 1):15–20.[CrossRef][ISI]
- Nyman B. On the one-dimensional translation group and semi-group in certain function spaces. Thesis, University of Uppsala, 1950.
- Picon P. A. Conditions d'intégrit'e de certains coefficients hypergéometriques: Généralisation d'un théorème de Landau" [Integrality conditions for certain hypergeometric quotients: generalization of a theorem of Landau]. Discrete Mathematics (1994) 135(no. 1–3):245–63.[CrossRef][ISI]
- Picon P. A. A more precise formulation of a theorem of Landau in the linear case and some applications. European Journal of Combinatorics (1994) 15(no. 6):561–77.[CrossRef][ISI]
- Picon P. A. Sum-translation and symmetry operators and integrality of hypergeometric linear coefficients. International Journal of Algebra and Computation (1995) 5(no. 1):19–45.[CrossRef]
- Reid M. Decomposition of toric morphisms. Arithmetic and Geometry. 395–418. Vol. 2. Progress in Mathematics 36. Boston, MA: Birkhauser, 1983.
- Rodriguez-Villegas F. Integral ratios of factorials and algebraic hypergeometric functions. (2007) preprint http://front.math.ucdavis.edu/math.NT/0701362.
- Sankaran G. K. Stable quintiples and terminal quotient singularities. Mathematical Proceedings of the Cambridge Philosophical Society (1990) 107(no. 1):91–101.[ISI]
- Stanley R. P. Enumerative Combinatorics. vol. 2. With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics 62. Cambridge: Cambridge University Press, 1999.
- Vasyunin V. I. On a system of step functions. Journal of Mathematical Sciences (2002) 110(no. 5):2930–43.[CrossRef]
- White G. K. Lattice tetrahedra. Canadian Journal of Mathematics (1964) 16:389–96.[ISI]
CiteULike 
Connotea 
Del.icio.us What's this?
| ||||||||||||||||||||||||||||||||||||||||||||||||