Related papers: On the $p^{\lambda}$ problem

We investigate the distribution of $p\theta$ modulo 1, where $\theta$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.

In this paper we analyze the asymptotic behaviour as $p\to 1^+$ of solutions $u_p$ to $$ \left\{ \begin{array}{rclr} -\Delta_pu&=&\lambda|\nabla u|^{p-2}\nabla u\cdot\frac{x}{|x|^2}+ f&\quad \mbox{ in } \Omega,\\ u_p&=&0 &\quad \mbox{ on…

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…

Let $\Psi$ be a system of linear forms with finite complexity. In their seminal paper, Green and Tao showed the following prime number theorem for values of the system $\Psi$: $$\sum_{x\in [-N,N]^d} \prod_{i=1}^t…

The intermediate dimensions of a set $\Lambda$, elsewhere denoted by $\dim_{\theta}\Lambda$, interpolates between its Hausdorff and box dimensions using the parameter $\theta\in[0,1]$. Determining a precise formula for…

Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if…

Let s_q(n) denote the base q sum of digits function, which for n<x, is centered around (q-1)/2 log_q x. In Drmota, Mauduit and Rivat's 2009 paper, they look at sum of digits of prime numbers, and provide asymptotics for the size of the set…

Let $\theta_0,\theta_1 \in \mathbb{R}^d$ be the population risk minimizers associated to some loss $\ell:\mathbb{R}^d\times \mathcal{Z}\to\mathbb{R}$ and two distributions $\mathbb{P}_0,\mathbb{P}_1$ on $\mathcal{Z}$. The models…

We study the question of whether for each n there is another integer m with lambda(m)=lambda(n), where lambda is Carmichael's function. We give a "near" proof of the fact that this is the case unconditionally, and a complete conditional…

We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta$ for $\theta=1/2+1/2000.$ This improves the work of Matom\"aki (2009) who obtained the result for $\theta=1/2-\varepsilon$ (with…

Hypothesis testing results often rely on simple, yet important assumptions about the behaviour of the distribution of p-values under the null and the alternative. We examine tests for one dimensional parameters of interest that converge to…

Previously, mathematicians Steven Krantz and Jeffery McNeal studied a type of positive numbers permutation called $\lambda$-permutation. This type of permutation, when applied to the index of terms of a series, is defined to be both…

The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ \Delta w-\frac{1}{2} \langle x,\nabla w\rangle-\frac{\lambda}{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical…

For a triangle $\Delta$, let (P) denote the problem of the existence of points in the plane of $\Delta$, that are at rational distance to the 3 vertices of $\Delta$. Answer to (P) is known to be positive in the following situation: $\Delta$…

Let $\Omega\subset\mathbb{R}^{n}$ be a smooth bounded domain and $m\in C(\overline{\Omega})$ be a sign-changing weight function. For $1<p<\infty$, consider the eigenvalue problem $$ \left\{ \begin{array} [c]{ll} -\Delta_{p}u=\lambda…

Let $P(n)$ be the number of polyominoes of $n$ cells and $\lambda$ be Klarner's constant, that is, $\lambda=\lim_{n\to\infty} \sqrt[n]{P(n)}$. We show that there exist some positive numbers $A,T$, so that for every $n$ \[ P(n) \ge…

In this paper, we consider the system $-\Delta u =\lambda (v+1)^p,\;\;-\Delta v = \gamma (u+1)^\theta$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$ with the Dirichlet boundary condition $u=v=0$ on $\partial \Omega.$ Here $…

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive…

This article study the fractional Hamiltonian systems \begin{eqnarray}\label{00} {_{t}}D_{\infty}^{\alpha}({_{-\infty}}D_{t}^{\alpha}u) + \lambda L(t)u = \nabla W(t, u), \;\;t\in \mathbb{R}, \end{eqnarray} where $\alpha \in (1/2, 1)$,…

Given an ordinal delta <= lambda and a cardinal theta <= kappa, an ideal J on P_kappa(lambda) is said to be [delta]^{<theta}-normal if given B_e in J for e in P_theta(delta), the set of all a in P_kappa(lambda) such that a in B_e for some e…