• Title, Summary, Keyword: zeta function

Search Result 180, Processing Time 0.035 seconds

SEVERAL RESULTS ASSOCIATED WITH THE RIEMANN ZETA FUNCTION

  • Choi, Junesang
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.22 no.3
    • /
    • pp.467-480
    • /
    • 2009
  • In 1859, Bernhard Riemann, in his epoch-making memoir, extended the Euler zeta function $\zeta$(s) (s > 1; $s{\in}\mathbb{R}$) to the Riemann zeta function $\zeta$(s) ($\Re$(s) > 1; $s{\in}\mathbb{C}$) to investigate the pattern of the primes. Sine the time of Euler and then Riemann, the Riemann zeta function $\zeta$(s) has involved and appeared in a variety of mathematical research subjects as well as the function itself has been being broadly and deeply researched. Among those things, we choose to make a further investigation of the following subjects: Evaluation of $\zeta$(2k) ($k {\in}\mathbb{N}$); Approximate functional equations for $\zeta$(s); Series involving the Riemann zeta function.

  • PDF

ZETA FUNCTIONS FOR ONE-DIMENSIONAL GENERALIZED SOLENOIDS

  • Yi, In-Hyeop
    • The Pure and Applied Mathematics
    • /
    • v.18 no.2
    • /
    • pp.141-155
    • /
    • 2011
  • We compute zeta functions of 1-solenoids. When our 1-solenoid is nonorientable, we compute Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid and its orientable double cover explicitly in terms of adjacency matrices and branch points. And we show that Artin-Mazur zeta function of orientable double cover is a rational function and a quotient of Artin-Mazur zeta function and Lefschetz zeta function of the 1-solenoid.

DETERMINANTS OF THE LAPLACIANS ON THE n-DIMENSIONAL UNIT SPHERE Sn (n = 8, 9)

  • Choi, June-Sang
    • Honam Mathematical Journal
    • /
    • v.33 no.3
    • /
    • pp.321-333
    • /
    • 2011
  • During the last three decades, the problem of evaluation of the determinants of the Laplacians on Riemann manifolds has received considerable attention by many authors. The functional determinant for the n-dimensional sphere $S^n$ with the standard metric has been computed in several ways. Here we aim at computing the determinants of the Laplacians on $S^n$ (n = 8, 9) by mainly using ceratin known closed-form evaluations of series involving Zeta function.

THE ZETA-DETERMINANTS OF HARMONIC OSCILLATORS ON R2

  • Kim, Kyounghwa
    • Korean Journal of Mathematics
    • /
    • v.19 no.2
    • /
    • pp.129-147
    • /
    • 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.

EVALUATIONS OF $\zeta(2n)$

  • Choi, June-Sang
    • East Asian mathematical journal
    • /
    • v.16 no.2
    • /
    • pp.233-237
    • /
    • 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.

  • PDF

ON SCATTERING BY SEVERAL OCNVEX BODIES

  • Ikawa, Mitsuru
    • Journal of the Korean Mathematical Society
    • /
    • v.37 no.6
    • /
    • pp.991-1005
    • /
    • 2000
  • We consider a zeta function of the classical dynamics in the exterior of several convex bodies. The main result is that the poles of the zeta function cannot converge to the line of absolute convergence if the abscissa of absolute convergence of the zeta function is positive.

  • PDF

REIDEMEISTER ZETA FUNCTION FOR GROUP EXTENSIONS

  • Wong, Peter
    • Journal of the Korean Mathematical Society
    • /
    • v.38 no.6
    • /
    • pp.1107-1116
    • /
    • 2001
  • In this paper, we study the rationality of the Reidemeister zeta function of an endomorphism of a group extension. As an application, we give sufficient conditions for the rationality of the Reidemeister and the Nielsen zeta functions of selfmaps on an exponential solvmanifold or an infra-nilmanifold or the coset space of a compact connected Lie group by a finite subgroup.

  • PDF