Related papers: A congruence with the Euler totient function

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results…

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's…

We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures.

In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).

Motivated by existing results, we present some completely monotonic functions involving the polygamma functions.

The completeness of Gaussians in a Hilbert functional space is established

In this article we construct a smooth Euler flow supported in a neighborhood of a helix. It may be considered a generalization of a similar solution found by the author for a circle.

We prove that suitable properties of the twists by Dirichlet characters of an L-function of degree 2 imply that its Euler product is of polynomial type.

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} \zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions,…

A formal theory of experimentation will be presented. Such a theory presents the necessary & sufficient conditions a world must satisfy in order to admit the use of the scientific method.

We have proposed a regularization technique and apply it to the Euler product of zeta functions in the part one. In this paper that is the second part of the trilogy, we give another evidence to demonstrate the Riemann hypotheses by using…

We present some completely monotonic functions involving the$q$-polygamma functions, our result generalizes some known results.

An existence and uniqueness theorem for a class of stochastic delay differential equations is presented, and the convergence of Euler approximations for these equations is proved under general conditions. Moreover, the rate of almost sure…

In this paper we improve a result on the order of magnitude of certain cotangent sums associated to the Estermann and the Riemann zeta functions.

We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.

In this paper, we consider a Class of Hessian quotient equations in Euclidean space. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the…

In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.

A generalization of the law of total covariance is presented and proved.

In this paper, we define extended trigonometric functions via series and employ the method of contour integration to investigate the parity of certain cyclotomic Euler sums and multiple polylogarithm function. We can provide the statement…

``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give…