相关论文: Elementary evaluations of some Euler sums

The main purpose of this Note is to provide a non-trivial bound on certain Kloosterman sums as considered in the recent paper of E. Fouvry [F], leading to a small improvement of his result on the Pell equation.

In this article we give a result obtained of an experimental way for the Euler totient function.

In this didactic note, we describe a procedure to derive successive approximations of $\pi$ using Euler Beta functions. It is an interesting exercise for undergraduate students, since it involves polynomial roots, integral calculations,…

In this short note we will use the residue theorem to establish a formula for Euler's constant. In particular, we offer a slightly generalized version of an interesting infinite series due to Flajolet, Gourdon, and Dumas.

Using Eulerian and Euler numbers, we establish congruences concerning sums involving harmonic numbers, tangent numbers and Genocchi numbers.

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones

Notes to lectures on the epsilon calculus, covering axioms, semantics, completeness, and the first epsilon theorem.

In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.

These are the notes from my courses on the arithmetic of quadratic forms.

In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications…

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.

We examine a family of three-dimensional exponential sums with monomials and provide estimates which are in some instances sharper than those stemming from approaches entailing the use of existing bounds pertaining to analogous sums.

We survey the classical results of the Dirichlet Approximation Theorem.

This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in…

Recently, Dil and Boyadzhiev \cite{AD2015} proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence $\left( {{{\left\{ 0 \right\}}_r},1} \right)$. In this paper we show that the sums of multiple…

In this paper we will investigate properties of modified q-Euler numbers and polynomials. The main purpose of this paper is to construct p-adic q-Euler measures.

The goal of this note is to provide a constructive version of the proof of local structure of etale algebras.

The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able…

The aim of this note is to show that Poincar\'e inequalities imply corresponding weighted versions in a quite general setting. Fractional Poincar\'e inequalities are considered, too. The proof is short and does not involve covering…