Related papers: The M\"obius function of the small Ree groups

For a connected semisimple real Lie group $G$ of non-compact type, Wallach introduced a class of $K$-types called small. We classify all small $K$-types for all simple Lie groups and prove except just one case that each elementary spherical…

In this paper, we will study the M\"obius polynomial, an invariant of ranked posets that arises in the study of splitting algebras. We will present a formula for the M\"obius polynomial of the direct product of posets in terms of the…

We prove that $d(G) \log |G| = O(n^2 \log q)$ for irreducible subgroups $G$ of GL$(n,q)$, and estimate the associated constants. The result is motivated by attempts to bound the complexity of computing the automorphism groups of various…

We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the M\"obius function of the Rees product of any graded poset with the chain, and more…

M\"obius transformations of the extended complex plane are at the crossroads of many interesting topics, e.g., they form a group under composition, are the simplest form of rational function, and are a path to Lie theory. Quaternionic…

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as…

We show that $$ \sum_{n\neq m}\frac{\mu(n)\mu(m)}{nm}E_{X}\left(\{nx\}\{mx\}\right)=-\frac{9}{2\pi^{2}}+O\left(\frac{1}{X}\right), $$ where $x$ is uniformly distributed in $[0,X]$ with $X\in \mathbb{N}$, $E_{X}(.)$ denotes the expected…

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a…

Let $\lambda$ and $\mu$ denote the Liouville and M\"obius functions respectively. Hildebrand showed that all eight possible sign patterns for $(\lambda(n), \lambda(n+1), \lambda(n+2))$ occur infinitely often. By using the recent result of…

The aim of this paper is to give a generalization of the theory equivariant functions, initiated in [17, 4], to arbitrary subgroups of PSL2(R). We show that there is a deep relation between the geometry of these groups and some analytic and…

In [Frobenius1896] it was shown that many important properties of a finite group could be examined using formulas involving the character ratios of group elements, i.e., the trace of the element acting in a given irreducible representation,…

A multiplicative function $f$ is said to be resembling the M\"{o}bius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $\Omega$-results for the summatory function $\sum_{n\leq…

Let $S(n)$ denote the least primary factor in the primary decomposition of the multiplicative group $M_n = (\Bbb Z/n\Bbb Z)^\times$. We give an asymptotic formula, with order of magnitude $x/(\log x)^{1/2}$, for the counting function of…

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function.…

Let $\Gamma$ be a totally ordered group. We use Hahn's embedding theorem to construct a totally ordered set $\Gamma\subset \Gamma_{\operatorname{sme}}$ which classifies small extensions of $\Gamma$. This small-extensions closure…

Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of…

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof…

Let $G(q)$ be a finite group of Lie type over a field with $q$ elements, where $q$ is a prime power. The Green functions of $G(q)$, as defined by Deligne and Lusztig, are known in \textit{almost} all cases by work of Beynon--Spaltenstein,…

We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we…

The article presents several methods for the arithmetic of finite abelian groups. We introduce a tool - already used by Delsarte in [1] as I found out later - analogous to Dirichlet's convolution to obtain combinatorial results on these…