• Honam Mathematical Journal
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• v.36 no.2
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• pp.263-273
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• 2014

Ever since Euler solved the so-called Basler problem of ${\zeta}(2)=\sum_{n=1}^{\infty}1/n^2$, numerous evaluations of ${\zeta}(2n)$ ($n{\in}\mathbb{N}$) as well as ${\zeta}(2)$ have been presented. Very recently, Ritelli [61] used a double integral to evaluate ${\zeta}(2)$. Modifying mainly Ritelli's double integral, here, we aim at evaluating certain interesting alternating series.

• East Asian mathematical journal
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• v.29 no.5
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• pp.545-550
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• 2013

A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1255-1280
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• 2017

In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. [2]. We then apply it to obtain a family of identities relating quadratic and cubic sums to linear sums and zeta values. Furthermore, we also evaluate several other series involving harmonic numbers and alternating harmonic numbers, and give explicit formulas.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.545-555
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• 2004

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a proof of the classical Euler sum by following Lewin's method. We also consider some related formulas involving Euler sums, which are evaluatable from some known definite integrals.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.137-146
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• 2013

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function.

• Honam Mathematical Journal
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• v.36 no.2
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• pp.467-472
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• 2014

With the properties of Riemann zeta function, we find an asymptotic formula of the sum for the absolute value of M$\ddot{o}$bius function, $\sum_{n{\leq}x}{\mid}{\mu}(n){\mid}$, using Dirichlet inversion formula.

• Journal for History of Mathematics
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• v.20 no.4
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• pp.71-84
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• 2007

J. Bernoulli first discovered the method which one can produce those formulae for the sum $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$ for any natural numbers k. After then, there has been increasing interest in Bernoulli and Euler numbers associated with Riemann zeta functions. Recently, Kim have been studied extended q-Bernoulli numbers and q-Euler numbers associated with p-adic q-integral on $\mathbb{Z}_p$, and sums of powers of consecutive q-integers, etc. In this paper, we investigate for the historical background and evolution process of the sums of powers of consecutive q-integers and discuss for Euler zeta functions subjects which are studying related to these areas in the recent.

• East Asian mathematical journal
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• v.18 no.1
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• pp.75-84
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• 2002

The third important Euler-Mascheroni constant $\gamma$, like $\pi$ and e, is involved in representations, evaluations, and purely relationships among other mathematical constants and functions, in various ways. The main object of this note is to summarize some known series representaions for $\gamma$ with comments for their proofs, and to point out that one of those series representaions for $\gamma$ seems to be incorrectly recorded. A brief historical comment for $\gamma$ is also provided.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1535-1549
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• 2020

This paper deals with the inverse of tails of Hurwitz zeta function. More precisely, for any positive integer s ≥ 2 and 0 ≤ a < 1, we give an algorithm for finding a simple form of f_s,a(n) such that $$\lim_{n{\rightarrow}{\infty}}\{\({\sum\limits_{k=n}^{\infty}}{\frac{1}{(k+a)^s}}\)^{-1}-f_{s,a}(n)\}=0$$. We show that f_s,a(n) is a polynomial in n-a of order s-1. All coefficients of f_s,a(n) are represented in terms of Bernoulli numbers.

• East Asian mathematical journal
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• v.31 no.1
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• pp.41-53
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• 2015

A large number of series and integral representations for the Stieltjes constants (or generalized Euler-Mascheroni constants) ${\gamma}_k$ and the generalized Stieltjes constants ${\gamma}_k(a)$ have been investigated. Here we aim at presenting certain integral representations for the generalized Stieltjes constants ${\gamma}_k(a)$ by choosing to use four known integral representations for the generalized zeta function ${\zeta}(s,a)$. As a by-product, our main results are easily seen to specialize to yield those corresponding integral representations for the Stieltjes constants ${\gamma}_k$. Some relevant connections of certain special cases of our results presented here with those in earlier works are also pointed out.