Related papers: On The Generalized Multiplicative Euler Phi Functi…

We prove simple theorems concerning the maximal order of a large class of multiplicative functions. As an application, we determine the maximal orders of certain functions of the type $\sigma_A(n)= \sum_{d\in A(n)} d$, where A(n) is a…

About a century ago, P. A. MacMahon introduced a class of $q$-series, which are nowadays referred to as MacMahon series. More recently, in 2013, G. E. Andrews and S. C. F. Rose revealed the quasimodular property of these series. In this…

Lehmer's totient problem asks whether there exists any composite number $n$ such that $\varphi(n) \, \mid \, (n-1)$, where $\varphi$ is Euler totient function. It is known that if any such $n$ exists, it must be Carmichael and $n >…

The present note generalizes a well-known formula for pi/2 named after the English mathematician John Wallis. The two new formulas for infinite products containing the natural numbers and their roots express them using the Euler-Mascheroni…

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields.…

We study a general class of Euler equations driven by a forcing with a \emph{commutator structure} of the form $[\mathcal{L},\mathbf{u}](\rho)=\mathcal{L}(\rho \mathbf{u})- \mathcal{L}(\rho)\mathbf{u}$, where $\mathbf{u}$ is the velocity…

Let $R = Z[C]$ be the integral group ring of a finite cyclic group $C$. Dennis and al. proved that $R$ is a generalized Euclidean ring in the sense of P. M. Cohn, i.e., $SL_n(R)$ is generated by the elementary matrices for all $n$. We prove…

The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q-exponential functions by one more parameter. we…

For an $n\times n$ bounded matrix function $\Phi$ we study unitary interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which the Toeplitz operator…

We study the integrable system of first order differential equations $\omega_i(v)'=\alpha_i\,\prod_{j\neq i}\omega_j(v)$, $(1\!\leq i, j\leq\! N)$ as an initial value problem, with real coefficients $\alpha_i$ and initial conditions…

Euclidean functions with values in an arbitrary well-ordered set were first considered in a 1949 work of Motzkin and studied in more detail in work of Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are revisited,…

The $ (k, a) $-generalized Fourier transform $ \mathscr{F}_{k, a} $ introduced by Ben Sa\"id--Kobayashi--{\O}rsted is a deformation family of the classical Fourier transform with a Dunkl parameter $ k $ and a parameter $ a > 0 $ that…

For a compact subset $K$ of the complex plane $\mathbb C,$ let $C(K)$ denote the algebra of continuous functions on $K$. For an open subset $U \subset K,$ let $A(K,U) \subset C(K)$ be the algebra of functions that are analytic in $U.$ We…

We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, labelled by rational cones in $\mathbb{R}^r$. For $r=2,3$ we prove that the generalized multiple elliptic gamma functions enjoy a modular…

In 1929 Heisenberg and Pauli laid the foundations of QFT by quantizing the fields (method of canonical quantization). This general theory of quantized fields has remained undisputed up to now. We show how the unmodified Heisenberg-Pauli…

For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m…

The Poincare series of a multi-index filtration on the ring of germs of functions can be written as a certain integral with respect to the Euler characteristic over the projectivization of the ring. Here this integral is considered with…

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$…

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]}$$ where $[x]$…

In recent, H. Sun defined a new kind of refined Eulerian polynomials, namely, \begin{eqnarray*} A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)} \end{eqnarray*} for $n\geq 1$, where ${odes}(\pi)$ and ${edes}(\pi)$…