Related papers: On the Composition of the Euler Function and the D…

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of…

We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with…

For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|\{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}\}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We improve the current…

We obtained order estimations for the best uniform approximations by trigonometric polynomials and approximations by Fourier sums of classes of $2\pi$-periodic continuous functions, which $(\psi,\beta)$-derivatives $f_{\beta}^{\psi}$ belong…

We prove that for any proper metric space $X$ and a function $\psi:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_\psi(Q)$ of all points $\psi$-well-approximable by a…

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $\varphi(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient…

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…

We introduce axiomatically the ring $\bf{Z}_\kappa$ of the Euclidean integers, that can be viewed as the ``integral part" of the field $\mathbb{E}$ of Euclidean numbers of [4], where the transfinite sum of ordinal indexed $\kappa$-sequences…

Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) = \sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask…

Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.

All the known approximations of the number of primes pi(n) not exceeding any given integer n are derived from real-valued functions that are asymptotic to pi(x), such as x/log x, Li(x) and Riemann's function R(x). The degree of…

We consider elliptic operators in divergence form with lower order terms of the form $Lu=-$div$\nabla u+bu)-c\nabla u-du$, in an open set $\Omega\subset \mathbb{R}^n$, $n\geq 3$, with possibly infinite Lebesgue measure. We assume that the…

Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of…

Euler showed that the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. This theorem was generalized by Glaisher and further by Franklin. Recently, Beck made three conjectures on…

Let $p_n(y)=\sum_k\hat{\alpha}_k\phi(y-k)+\sum_{l=0}^{j_n-1}\sum_k\hat {\beta}_{lk}2^{l/2}\psi(2^ly-k)$ be the linear wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support),…

We consider the problem of estimating the density $\Pi$ of a determinantal process $N$ from the observation of $n$ independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish…

In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X)…

Let $A$ be the product of an abelian variety and a torus defined over a number field $K$. Fix some prime number $\ell$. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the…

We propose a lower estimation for computing quantity of the inverses of Euler's function. We answer the question about the multiplicity of $m$ in the equation $\varphi(x) = m$ \cite{Ford}. An analytic expression for exact multiplicity of $m…

Dykema, Sukochev and Zanin used a Peano curve covering the support of the Brown measure of an operator T in a diffuse, finite von Neumann algebra to give an ordering to the support of the Brown measure, and create a decomposition T = N + Q,…