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\title{$\xi$ - $\zeta$ relation}
\author{M. Krishna\thanks
{E-mail: krishna@imsc.ernet.in}\\
Institute of Mathematical Sciences\\ Taramani,
Chennai 600 113, India}
\date{}
\begin{document}
\maketitle
\begin{abstract}
In this note we prove a relation between the Riemann Zeta function,
$\zeta$ and the $\xi$ function (Krein spectral shift)
associated with the Harmonic Oscillator in one dimension. This gives
a new integral representation of the zeta function.
\end{abstract}
\newpage
\section{Introduction}
The Riemann zeta function is a well studied object, for example,
Titchmarch \cite{tit} gives a detailed exposition of this function.
There are several expressions for $\zeta$, and in this note we present
an integral representation for $\zeta$, that comes from the Krein
spectral shift formula of Krein \cite{kr1, kr2}. Recently Gesztesy-Simon \cite{gs}
generalized the trace formulae for Schr\"odinger
operators using the Krein spectral shift function, which they named the $\xi$
function, as it is central to inverse spectral theories in one
dimension and had several important applications in spectral theories
of operators in one dimension.
This work used the proof of the
Krein formula, given in Simon \cite{si}, theorem I.10 and its
generalizations.
A proof of the formula for a slightly larger
class is shown in Mohapatra-Sinha \cite{ms}. We refer to these
papers for the history and other work on the Krein spectral shift
function.
\section{$\xi$ function of the Harmonic oscillator}
In this section we recall the $\xi$ function of
the Harmonic oscillator.
We consider the Harmonic oscillator,
H = $\half(-\frac{d^2}{dx^2} + x^2 + 1)$, acting on $L^2(\RR)$,
normalized so that its spectrum is the
positive integers $\ZZ^+$. We consider the operator $H_{\infty}$,
using the notation of \cite{gs, si},
obtained by looking at the Harmonic oscillator with Dirichlet
condition at 0. Then the spectrum of $H_{\infty}$ is even integers
$2\ZZ^+$, with uniform multiplicity 2. The Krein
spectral shift function $\xi(\lambda)$ for the pair of operators (H, $H_{\infty}$) is given
by
$$
\xi(\lambda) = \sum_{m=1}^{\infty} \chi_{[2m-1, 2m)}(\lambda)
$$
where $\chi_X$ denotes the indicator function of X. In terms of the
$\xi$ function, Gesztesy-Simon \cite{gs}, Simon \cite{si} theorem I.10
(case $\alpha = \infty$), and
Mohapatra-Sinha theorem 4.2 , proved the trace formula,
\begin{equation}
Tr(f(H) - f(H_{\infty})) = -\int (f(\lambda))^{\prime} \xi(\lambda) d\lambda
\label{trace}
\end{equation}
with different smoothness and decay conditions on f.
Fix $s=\sigma + it$ and consider the smooth function
$f(\lambda) = \lambda^s$, on [1,$\infty$),
then by functional calculus, it follows that,
$$
f(H) = \int f(\lambda) dE_H(\lambda), ~~ and ~~ f(H_{\infty}) = \int f(\lambda)
dE_{H_{\infty}}(\lambda)
$$
are both trace class for $\sigma > 2$, since
$$
\sum_{m = 1}^{\infty} m^{-\sigma} ~~ and ~~ \sum_{m=1}^{\infty}
(2m)^{-\sigma}
$$
converge.
Here is the primary relation between the $\xi$ and $\zeta$ functions.
\begin{thm}
Let $\xi(\lambda)$ denote the Krein spectral shift function for the pair of
operators (H, $H_{\infty}$), defined above. Then the Riemann zeta function
$\zeta$(s) is related to $\xi$ through the relation,
\begin{equation}
(1 - 2^{(1-s)})\zeta(s) = s\int_1^{\infty} \lambda^{-s-1} ~~ \xi(\lambda) d\lambda
\label{xizeta}
\end{equation}
valid for any $s =\sigma + it$, with $\sigma > 0$.
\end{thm}
\proof.
We consider $s = \sigma + it$ with $\sigma > 2$, then we take f(x) =
$x^{(-s)}$. Then by definition $\zeta(s) = Trace(f(H))$. Now we rewrite
this as,
$$
\zeta(s) = Trace(f(H) - f(H_{\infty})) + Trace(f(H_{\infty})
$$
by the linearity of the trace. Now we notice that
since the spectral multiplicity of $H_{\infty}$ is 2 and the spectrum is the
even integers we have,
$$
Trace(f(H_{\infty})) = 2\sum_{n=1}^{\infty} (2n)^{(-s)} = 2^{(1-s)}\sum_{n=1}^{\infty}
(n)^{(-s)} = 2^{(1-s)}\zeta(s).
$$
Using the above relations, and the trace formula in terms of the Krein
spectral shift, we immediately see that for $\sigma >2$,
the theorem is valid, since then the function f($\lambda$) = ${\lambda}^{-s}$
is $C^2$ and satisfies $(1 + |\lambda|^2)f^{j}(\lambda) \in L^2(\RR^+),
j=1,2$,
while its extension to $\sigma >0$ follows from
the analyticity of the left and right hand sides of equation
\ref{xizeta}.
A simple change of variables $\ln \lambda = x$ in the expression for $\zeta$ given in
the above theorem gives the following corollary.
In the following we take,
$$
\phi(x) = \sum_{n = 1}^{\infty} \chi_{[\ln(2n-1), \ln(2n))}(x).
$$
\begin{cor}
The zeta function is given, in the region $\sigma > 0$, $s = \sigma +
it$, by,
$$
\zeta(s) = \frac{s}{(1 - 2^{1-s})}\int_{0}^{\infty} e^{-s x} \phi(x) dx
$$
\end{cor}
\begin{rem}
In the context of inverse spectral theory of the Harmonic oscillator,
we could have taken instead of $H_{\infty}$ with boundary condition at 0, the
operators with the Dirichlet condition at any x, however this will not
lead to a closed expression for the $\zeta$ function. It may still
happen for special non-zero values of x (those for which all but
finitely many of the Dirichlet eigen values again fall in the spectrum of H)
and it would be interesting to identify these values of x and the
resulting identities for $\zeta$, if they exist.
\end{rem}
\begin{thebibliography}{99}
\bi{gs} F. Gesztesy and B. Simon, {\em The $\xi$ function},
Acta. Mathematica {\bf 176}, 46-71 (1996).
\bi{ms} Kalyan B. Sinha and A. N. Mohapatra, {\em Spectral shift
function and trace formula}, Proc. Ind. Acad. Sci.(Math.
Sci.),{\bf 104}(4), 819-853(1995).
\bi{kr1}M. G. Krein, {\em On perturbation determinants and a trace
formula for unitary and self-adjoint operators}, Soviet Math. Doklady,
{\bf 3}, 707-710(1963).
\bi{kr2}M. G. Krein, in {\em Topics in integral and
differential equations and operator theory}, (Ed. I. Gohberg),
Birkhauser-Verlag, Basel (1983).
\bi{si} B. Simon, {\em Spectral analysis of rank one
perturbations and applications}, Mathematical Quantum theory
II: Schr\"odinger operators, (eds.,) J. Feldman, R. Froese and
L.M. Rosen , CRM Proceedings, American Mathematical Society,
Providence, RI (1995).
\bibitem{tit} E. C. Titchmarch, {\em The theory of the Riemann Zeta
function}, Oxford University Press, Oxford 1951.
\end{thebibliography}
\end{document}