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

TL;DR

This paper classifies a family of step functions linked to the Riemann hypothesis, revealing when factorial ratios are integral, using hypergeometric function monodromy classification, with applications to singularity classification.

Contribution

It provides a complete classification of step functions related to factorial ratios and hypergeometric series, connecting number theory and algebraic geometry.

Findings

Characterization of when factorial ratios are integral

Connection to hypergeometric function monodromy classification

Applications to cyclic quotient singularities

Abstract

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 V. I. Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations are made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nF_{n-1}. We also note applications to the classification of cyclic quotient singularities.