相关论文: Elementary evaluations of some Euler sums

This paper evaluates some generalised Euler sums involving the digamma function.

This note contains some asymptotic formulas for the sums of various residue classes of Euler's phi-function.

In this short note, we obtain error estimates for Riemann sums of some singular functions.

These informal notes are concerned with sums and averages in various situations in analysis.

Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.

These notes give an elementary approach to parts of the theory of standard Borel and analytic spaces.

These informal notes deal with a number of questions related to sums and integrals in analysis.

The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between…

This note highlights an interesting connection between Euler sums of even weight and prime numbers.

This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.

Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including…

We give a complete and elementary proofs of "Jordan's sums" and study Euler's types sums. In particular we give a formula for the sum of series with same weight, which is similar to this one of classical 2-Euler's sums.

In this paper, we derive some interesting symmetric properties for the geenralized Euler numbers and polynomials.

In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.

This is a short survey of amenable equivalence relations.

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of…

We aim to investigate the four types of variant Euler harmonic sums. Also, as corollaries, we provide particular examples of our core findings, some of whose further instances are evaluated in terms of basic and well-known functions as well…

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 $[x]$…

We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a…