Related papers: The investigation of Euler's totient function prei…

This survey of alternating permutations and Euler numbers includes refinements of Euler numbers, other occurrences of Euler numbers, longest alternating subsequences, umbral enumeration of classes of alternating permutations, and the…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

We consider the expansion of an integral considered by F.G. Tricomi given by \[\int_{-\infty}^\infty x e^{-x^2}(\frac{1}{2}+\frac{1}{2}\mbox{erf}\,x)^{m} dx\] as $m\to\infty$. The procedure involves a suitable change of variable and the…

We investigate bounds for the multiplicities $m(\beta+i\gamma)$, where $\beta+i\gamma\,$ ($\beta\ge \1/2, \gamma>0)$ denotes complex zeros of $\zeta(s)$. It is seen that the problem can be reduced to the estimation of the integrals of the…

This paper concerns the values of the Euler phi-function evaluated simultaneously on k arithmetic progressions a_1 n + b_1, a_2 n + b_2, ..., a_k n + b_k. Assuming the necessary condition that no two of the polynomials a_i x + b_i are…

We characterize the possible behaviors at infinity of weak solutions to the 2D Euler equations in the full plane having bounded velocity and bounded vorticity. We show that any such solution can be put in the form obtained by Ph. Serfati in…

Let $M(n)$ denote the number of distinct entries in the $n \times n$ multiplication table. The function $M(n)$ has been studied by Erd\H{o}s, Tenenbaum, Ford, and others, but the asymptotic behaviour of $M(n)$ as $n \to \infty$ is not known…

Let $M(\alpha)$ denote the Mahler measure of the algebraic number $\alpha$. In a recent paper, Dubickas and Smyth constructed a metric version of the Mahler measure on the multiplicative group of algebraic numbers. Later, Fili and the…

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 >…

Nicolas inequality we deal can be written as \begin{equation}\label{Nicineq} e^\gamma \log\log N_x < \dfrac{N_x}{\varphi(N_x)}\,, \end{equation} where $x\ge 2$, $N_x$ denotes the product of the primes less or equal than $x$, $\gamma$ is the…

We obtain exact and asymptotic expressions for the excess of nonnegative (2m+1)-multiples less than (2m)^k with even digit sums in the base 2m.

Flajolet and Salvy pointed out that every Euler sum is a $\mathbb{Q}$-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and…

We show that the solution of the Cauchy problem for the classical ode $m \mathbf y''=\mathbf f$ can be obtained as limit of minimizers of exponentially weighted convex variational integrals. This complements the known results about weighted…

We establish nontrivial bounds for bilinear sums involving the M\"obius function evaluated over solutions to a broad class of equations. Several of our results may be regarded as M\"obius-function analogues of the ternary Goldbach problem.…

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M. A. Stern.

In this paper, we define a parametric variant of generalized Euler sums and call them the (alternating) parametric Euler $T$-sums. By using the contour integration method and residue theorem, we establish several explicit formulae for the…

In this paper we propose a multiscale scanning method to determine active components of a quantity $f$ w.r.t. a dictionary $\mathcal{U}$ from observations $Y$ in an inverse regression model $Y=Tf+\xi$ with linear operator $T$ and general…

We consider different generalizations of the Euler formula and discuss the properties of the associated trigonometric functions. The problem is analyzed from different points of view and it is shown that it can be formulated in a natural…

The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of…

We present a new tool to compute the number $\phi_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are integral. $\phi_\A (\b)$ is often described as a \emph{vector…