• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.165-174
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• 2012

In this paper, we construct a p-adic analogue of multiple Riemann zeta values and express their values at non-positive integers. In particular, we obtain a new congruence of the higher order Frobenius-Euler numbers and the Kummer congruences for the Bernoulli numbers as a corollary.

• East Asian mathematical journal
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• v.16 no.2
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• pp.233-237
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• 2000

Since the time of Euler, there have been many proofs giving the value of $\zeta(2n)$. We also give an evaluation of $\zeta(2n)$ by analyzing the generating function of Bernoulli numbers.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.7A
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• pp.503-510
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• 2003

In CDMA cellular system, because all users share the frequency resource the signals of other user becomes interference which influences the communication quality. The system capacity defined the number of connected users within a cell is determined by the amount of interference, therefore the exact estimation of interference is important to system performance evaluation. In this paper, we propose an approximated function which calculates other cell interference in terms of Riemann-Zeta function in CDMA cellular systems, and compare with simulation results in other to verify its usefulness. The upper and lower bounds of system capacity calculated with the proposed approximated function gives almost alike result with the simulation. The proposed interference bounds are useful to calculate system capacity and to evaluate some algorithm in a hierarchical cellular systems where various propagation environments are mixed.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1671-1683
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• 2016

Abstract. In this paper, for $a{\in}C$, we investigate functions $g_a$ and ${\psi}_a$ associated with three term relations. $g_a$ is defined by means of function ${\psi}_a$. By using these functions, we obtain some functional equations related to the Eisenstein series and the Riemann zeta function. Also we find a generalized difference formula of function $g_a$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.29-41
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• 2014

In this paper, we prove that infinite-dimensional vector spaces of -dense curves are generated by means of the functional equations f(x)+f(2x)+${\cdots}$+f(nx) = 0, with $n{\geq}2$, which are related to the partial sums of the Riemann zeta function. These curves ${\alpha}$-densify a large class of compact sets of the plane for arbitrary small ${\alpha}$, extending the known result that this holds for the cases n = 2, 3. Finally, we prove the existence of a family of solutions of such functional equation which has the property of quadrature in the compact that densifies, that is, the product of the length of the curve by the $n^{th}$ power of the density approaches the Jordan content of the compact set which the curve densifies.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.129-147
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• 2011

In this paper we discuss the zeta-determinants of harmonic oscillators having general quadratic potentials defined on $\mathbb{R}^2$. By using change of variables we reduce the harmonic oscillators having general quadratic potentials to the standard harmonic oscillators and compute their spectra and eigenfunctions. We then discuss their zeta functions and zeta-determinants. In some special cases we compute the zeta-determinants of harmonic oscillators concretely by using the Riemann zeta function, Hurwitz zeta function and Gamma function.

• Honam Mathematical Journal
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• v.37 no.4
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• pp.423-429
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• 2015

Dirichlet series is a Riemann zeta function attached with an arithmetic function. Here, we studied the properties of Dirichlet series and found some identities between arithmetic functions.

• East Asian mathematical journal
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• v.16 no.2
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• pp.303-315
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• 2000

The authors apply the theory of multiple Gamma functions, which was recently revived in the study of the determinants of the Laplacians, in order to present a class of closed-form evaluations of series involving the Zeta function by appealing only to the definitions of the double and triple Gamma functions.

• East Asian mathematical journal
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• v.16 no.1
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• pp.1-7
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• 2000

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1163-1184
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• 2007

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called Basler problem of evaluating the Zeta function ${\zeta}(s)$ [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when s=2, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of ${\zeta}(s)$ when $s{\in}{\mathbb{N}}{\backslash}\;[1],\;{\mathbb{N}}$ being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for ${\zeta}(2n+1)(n{\in}{\mathbb{N}})$ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that ${\zeta}(3)$ can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger $Ap\'{e}ry$ (1916-1994) in his proof of the irrationality of ${\zeta}(3)$. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.