Quotient singularities, integer ratios of factorials and the Riemann Hypothesis

On Quotients of Values of Euler's Function on Factorials

Recently, there has been some interest in values of arithmetical functions on members of special sequences, such as Euler's totient function $ϕ$ on factorials, linear recurrences, etc. In this article, we investigate, for given positive integers $a$ and $b$, the least positive integer $c=c(a,b)$ such that the quotient $ϕ(c!)/ϕ(a!)ϕ(b!)$ is an integer. We derive results on the limit of the ratio $c(a,b)/(a+b)$ as $a$ and $b$ tend to infinity. Furthermore, we show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$ with an exception of a set of density zero.

Bounded step functions and factorial ratio sequences

We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities.

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

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.

On quotients of Riemann zeta values at odd and even integer arguments

We show for even positive integers $n$ that the quotient of the Riemann zeta values $ζ(n+1)$ and $ζ(n)$ satisfies the equation $$\frac{ζ(n+1)}{ζ(n)} = (1-\frac{1}{n}) (1-\frac{1}{2^{n+1}-1}) \frac{\mathcal{L}^\star(\mathfrak{p}_n)}{\mathfrak{p}_n'(0)},$$ where $\mathfrak{p}_n \in \mathbb{Z}[x]$ is a certain monic polynomial of degree $n$ and $\mathcal{L}^\star: \mathbb{C}[x] \to \mathbb{C}$ is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition $\mathfrak{p}_n(x) = x(x+1) \mathfrak{q}_n(x)$. If $n = p+1$ where $p$ is an odd prime, then $\math…

Integrality of hook ratios

We study integral ratios of hook products of quotient partitions. This question is motivated by an analogous question in number theory concerning integral factorial ratios. We prove an analogue of a theorem of Landau that already applied in the factorial case. Under the additional condition that the ratio has one more factor on the denominator than the numerator, we provide a complete classification. Ultimately this relies on Kneser's theorem in additive combinatorics.

通过相交理论与商奇点研究基变换导子

Base change conductors through intersection theory and quotient singularities

本文利用相交理论与 Deligne 的 Riemann–Roch 定理，系统建立 Jacobian 基变换导子驯顺与野生部分的新计算方法，并分析弱野生商奇点下的潜在好约化。