Takashi Ichinose and Masato Wakayama
On the Spectral Zeta Function for the Non-commutative 
Harmonic Oscillator
(35K, LaTeX 2e)

ABSTRACT.  The spectral zeta function for the so-called non-commutative 
harmonic oscillator 
is able to be meromorphically extended to the whole complex 
plane, having only one simple pole at the same point $s=1$ where 
Riemann's zeta function $\zeta(s)$ has, 
and possesses a trivial zero at each non-positive even integer. 
The essential part of its proof is sketched. A new result is also given 
on the lower and upper bounds of the eigenvalues of the non-commutative 
harmonic oscillator.