In 1970, Apostol introduced the Möbius function of order k for an integer $k\ge 1$. In 2001, it was generalized by Bege to the Möbius function $\mu _{k,m}$ of two parameters for any integers $m\ge k\ge 1$. In this article, we will establish three kinds of asymptotic formulas related to $\mu _{k,m}$ for $m\ge k\ge 2$ in a unified elementary method. These formulas are related to the shifted convolution sums of Fourier coefficients of holomorphic cusp forms, Bergelson and Richter’s dynamical generalization of the prime number theorem, and the Titchmarsh divisor problem.

We prove two results, generalizing certain theorems by Jin and Moosa (2020, Trans. Amer. Math. Soc., 373, 4863–4887), on the internality of the system of differential equations

$$ \begin{align*} \begin{cases} \begin{aligned} x' &= f(x)\\ y' &= g(x)y,\\ \end{aligned} \end{cases} \end{align*} $$

Let K be an absolutely abelian number field, and t be a positive integer relatively prime to $[K: \mathbb {Q}]$. We provide a method to use the t-rank of the class group of K to obtain a lower bound on the t-rank of class groups of any cyclotomic fields (or real cyclotomic fields) that contain this K. As an application, for any odd integer $t> 1$, we provide a family of cyclotomic fields whose class groups have t-rank at least $2$. Furthermore, we prove that the class number of $\mathbb {Q}(\zeta _{p^e} + \zeta _{p^e}^{-1})$ cannot be equal to $2$, where p is a prime, $e \geq 1$ is an integer, and $\zeta _{p^e}$ denotes a primitive $p^e$-th root of unity. We also provide an upper bound for the class number of a real cyclotomic field.

We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we exhibit two families of rational cubic fourfolds that are not equivariantly rational with respect to their group of automorphisms. As an application, we determine the cohomological action of symplectic birational transformations of manifolds of OG10 type that are induced by prime order symplectic automorphisms of cubic fourfolds.

Symmetrically self-similar graphs are an important type of fractal graph. Their Green’s functions satisfy order one iterative functional equations. We show that when the branching number of a generating cell is two, either the graph is a star consisting of finitely many one-sided lines meeting at an origin vertex, in which case the Green’s function is algebraic, or the Green’s function is differentially transcendental over $\mathbb {C}(z)$. The proof strategy relies on a result in a recent preprint of Di Vizio, Fernandes, and Mishna. The result adds evidence to a conjecture of Krön and Teufl about the spectra of the difference Laplacian of this family of graphs.

A pair of Dehn fillings on a compact, orientable 3-manifold M with a torus boundary $\partial M$ is said to be purely cosmetic if the resulting 3-manifolds are orientation-preservingly homeomorphic. In this article, we show that if $\partial M$ is incompressible, then there are only finitely many pairs of purely cosmetic fillings.

Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha $ is equal to $\pi /\sin ((2+\alpha )\pi /p)$ when $-1<\alpha <p-2$. In this article, we provide a proof of this conjecture for $0\leq \alpha \leq \frac {6p^3-29p^2+17p-2+2p\sqrt {6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha $ improves the best known result when $\alpha>\frac {1}{47}$ and $\alpha \not =1$.

We show that there are locally compact spaces that can be condensed on separable spaces, but not on compact separable spaces. We also show that for every cardinal $\kappa ,$ there is a locally compact topological group of cardinality $2^\kappa $ that can be condensed on a compact space but not on a compact topological group. These answer some questions of Arhangel’skii and Buzyakova.

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves with rational components in terms of the combinatorics of the corresponding graphs.

We develop several $\ell ^p$-operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$-sum of Banach spaces. Using these inequalities, we obtain p-numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p-numerical radius bound for the Kronecker product $A\otimes B$, where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$. This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$-operator norm and p-numerical radius of $A\in {M}_k(\mathbb {C})$, respectively, then it is shown that

$$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$

$\mu _p(A)$

$\ell ^p$

$\frac {1}{2}\|A\|_p= w_p(A)$

It is known that the condition $|\arg f'(z)|<\pi /2$, $|z|<1$, is not sufficient for an analytic function $f(z)=z+a_2z^2+\cdots $ in $|z|<1$, to be starlike with respect to the origin. We look for the largest $\alpha>0$ such that the condition $|\arg f'(z)|<\alpha \pi /2$ in $|z|<1$ is a sufficient condition for f to be a univalent, starlike, convex, or Bazilevic̆ function.

Let K be a genus one two-bridge knot. Let p be a prime number and let ${\mathbb {Z}}_{p}$ denote the ring of p-adic integers. In the spirit of arithmetic topology, we observe that if $p\neq 2$ and p divides (or $p=2$ and $2^3$ divides) the size of the 1st homology group of some odd-th cyclic branched cover of the knot K, then its group $\pi _1(S^3-K)$ admits a liminal $\mathrm { SL}_2{\mathbb {Z}}_p$-character. In addition, we discuss the existence of liminal $\mathrm {SL}_2{\mathbb {Z}}_{p}$-representations and give a remark on a general two-bridge knot. In the course of the argument, we also point out a constraint for prime numbers dividing certain Lucas-type sequences by using the Legendre symbols.

Given an action by a finite quantum group $\mathbb {G}$ on a von Neumann algebra M, we prove that a number of familiar $W^*$ properties are equivalent for M and the fixed-point algebra $M^{\mathbb {G}}$ (i.e., hold or not simultaneously for the two algebras); these include being hyperfinite, atomic, diffuse, and of type I, $II$, or $III$. Moreover, in all cases, the canonical central projections of M and $M^{\mathbb {G}}$ cutting out the summand with the respective property coincide. The result generalizes its classical-$\mathbb {G}$ analog due to Jones–Takesaki.

We examine “stronger” versions of the Rost nilpotence principle, which hold in almost all cases where Rost nilpotence is known, and show that a direct sum of motives for which these stronger principles hold also satisfy it.

We consider the family of infinite positive Borel measures $\mu $ in the unit disc, with finite degree of contact at the unit circle, and the set $E_{\mu }$ of points in the unit circle for which every neighborhood contains infinite mass of $\mu $. Assuming that $E_{\mu }$ is a Carleson set, we show that all Blaschke sequences are zero sets for the corresponding Dirichlet space $D_{\mu }$.

The Unitary Dual Problem is one of the most important open problems in mathematics: classify the irreducible unitary representations of a given group. It is known for a real reductive Lie group that $A_{\mathfrak {q}}(\lambda )$ modules are unitary and that any unitarizable Harish-Chandra module of strongly regular infinitesimal character is isomorphic to an $A_{\mathfrak {q}}(\lambda )$. Thus, it is of interest to study representations of singular infinitesimal character. For a compact real form and any alcove of the form $w(-\lambda + \underline {A}_\circ ),$ where $\lambda $ is dominant (possibly singular) and $\underline {A}_\circ $ is the dominant fundamental alcove, the signature character of the canonical invariant Hermitian form on the irreducible Verma module of infinitesimal character in that alcove is the “negative” of a Hall–Littlewood polynomial summand at $q=-1$ times a version of the Weyl denominator. (Signature characters for other real forms and alcoves of other forms may also be expressed using Hall–Littlewood polynomial summands.) Such formulas give hope that the Unitary Dual Problem is tractable in the singular case.

In this note, the author recalls the Calderon–Zygmund theory on the unit ball and derives the weak (1,1) boundedness of the projection for $\mathcal {H}$-harmonic Bergman space.

Let ${\mathbb {F}_q}$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod p representations of ${G_q} = \text {GL}_2({{\mathbb {F}_q}})$ to ${G_p} = \text {GL}_2({\mathbb {F}_p})$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup ${B_q}$. We then analyze the restrictions of inductions from an anisotropic torus ${T_q}$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of f. The proofs depend on writing down explicit orbit decompositions of ${G_p} \backslash {G_q} / H,$ where $H = {B_q}$ or ${T_q}$ using the fact that ${G_q}/H$ is an explicit orbit in a certain projective line, along with Mackey theory.

In this study, a Holling–Tanner type predator–prey model with a discrete time delay is investigated, where the functional response of the predator dynamics is ratio-dependent. We first analyze the local stability of the equilibrium point and examine the existence of Hopf bifurcations. The Hopf bifurcation, also known as the Poincaré–Andronov–Hopf bifurcation, is named after the French mathematician Jules Henri Poincaré, the Russian mathematician Alexander A. Andronov, and the German mathematician Heinz Hopf, whose fundamental contributions laid the foundation of this theory. By treating the delay parameter $\tau $ as the bifurcation parameter, we show that a Hopf bifurcation occurs when the delay crosses certain critical values. Finally, numerical simulations are carried out to support and illustrate our theoretical results.

In this article, certain classes of homeomorphisms on the complex plane have been considered. Specifically, we investigate ring and lower Q-homeomorphisms with respect to the p-module, as well as homeomorphisms of finite distortion. The behavior at infinity of these mappings is investigated, and sharp estimates for their lower power $\alpha $-order are obtained.