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\title{$\xi$ - $\zeta$ relation}



\author{M. Krishna\thanks
{Part of talk presented at the Conference on {\em Harmonic Analysis,
13-15 March 1997},
Ramanujan Institute, University of Madras, Chennai}\\
Institute of Mathematical Sciences\\ Taramani,
Chennai 600 113, India\\
E-mail: krishna@imsc.ernet.in}
\date{7 January 1999}
\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 and also a
reformulation of the Riemann hypothesis as a question in 
$L^1(\RR)$. 
\end{abstract}

\newpage

\section{Introduction}
Inverse spectral theory in one dimension involves recovering a
Schr\"odinger operator from the knowledge of spectrum and a spectral
function as done by Gelfand-Levitan in the fifties.  In the recent
years there is a great deal of progress achieved in parametrizing
iso-spectral classes of potentials (see the reviews of Simon
\cite{si}, Gesztesy \cite{ge}, and the papers of Levitan \cite{le},
Kotani-Krishna \cite{kk}, Craig \cite{cr} and Sodin-Yuditskii
\cite{sy} for Schr\"odinger operators).  One of the consequences of
a general formulation obtained via using the Krein spectral shift
function by Gesztesy-Simon \cite{gs} is given in this paper.

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.

Finally we note that the reformulation we obtain for the
Riemann hypothesis is a closure problem in the space $L^1(\RR)$.  
Though, via the powerful Wiener's theorem, the verification only
requires exhibiting a single function $g_\sigma$ (for each $\sigma \in (1/2 , 1)$)
to lie in an explicit subspace $X_\sigma$. 

A. Beurling provided an equivalent condition earlier in his paper
\cite{ab}, (also see Donoghue \cite{do}) which reformulates the 
Riemann hypothesis as a completeness problem in $L^2(0,1)$.
Lee Jungseob \cite{lj} gave another such reformulation. 


\section{$\xi$ function of the Harmonic oscillator} 

In this section we recall the $\xi$ function of
the Harmonic oscillator (from Gesztesy-Simon \cite{gs}).  

We consider the  Harmonic oscillator,
H = $\half(-\frac{d^2}{dx^2} + x^2 + 1)$, acting on 
$L^2(\RR)$ with the set $C_0^\infty(\RR)$ its domain of essential
self-adjointness and  
normalized so that its spectrum is the
positive integers $\ZZ^+$.  We consider the operator 
$H_{\infty, x}$ defined on $L^2((-\infty, x)) +  L^2((x, \infty))$
as H together with the Dirichlet condition $f(x) = 0$ at x,   
using the notation of \cite{gs, gs1, si}.
We denote by $H_\infty$ the operator $H_{\infty, 0}$in the following.
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,$\infty$), and zero in $(-\infty,
0)$,  
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}

\pf.
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}

\noindent {\bf Remark:} 1. Once the above relation is written down it
is trivial to prove directly from the definition of the $\zeta$
function, using integration by parts, but the connection with the
spectral problem is essential for the next part of the remark.


2. We could have considered the Dirichlet operator associated with any
point x $\in \RR$, in which case we can define a new family of
functions $\zeta(x, s)$, given by 
$$
\zeta(x, s) = Tr ( f(H) - f(H_{\infty, x})) = s \int_1^\infty
\frac{1}{\lambda^{s + 1}} \xi(\lambda, x) d\lambda 
$$
where $\xi(\lambda, x) = \sum_{n = 1}^\infty \chi_{[n,
\mu_n(x))}(\lambda) $ is the $\xi$ function associated with the pair
$H, H_{\infty, x}$ and as before we first define the sum on the right
hand side via the
integrals for $\sigma > 2$ and then extend them to $\sigma > 0$.
The expression for the right hand side agrees with the sum
Then the non-constant points of discontinuity of $\zeta(x, s)$ 
satisfy a differential
equation in x, called the Dubrovin equation (familiar in
inverse spectral theory, see \cite{cr}, \cite{gs}, \cite{kk}, \cite{sy}),
$$
\frac{d \mu_i(x)}{dx} = \sigma_i(x) \left( \frac{\partial
g_{\lambda}(x,x)}{\partial\lambda}\vline_{\lambda = \mu_i(x)}\right)^{-1}, ~~ i =
1, 2, 3, \cdots . 
$$
where $g_\lambda(x,x) = \lim_{\epsilon \rightarrow 0} G(\lambda +
i\epsilon, x, x)$, G being the Green function associated with H. 
This differential equation gives a curve in the space of analytic
functions on $\CC$ with the zeta function (times the factor $(1 -
2^{1-s})/s$) as the initial value.  An
explicit analysis of the equation should provide a new tool to study
the zeta function.


Once we have the above expression for $\zeta$, we can use the Wiener's
$L^1$ Tauberian theorem (see Wiener \cite{wi}, Section 14,
Theorem 9) to obtain the following reformulation of the  
the Riemann hypothesis. 

\begin{prop}\label{suff}
Consider the function
$f_{\sigma} (\lambda) = e^{-\sigma\lambda}\phi(\lambda)$.
Fix, $1 > \sigma > 1/2$.  Let $X_{f_\sigma}$ denote the subspace generated
by finite linear combinations of the translates of $f_\sigma$.
Then, $\widehat{f_\sigma}$ is zero free if and only if $X_{f_\sigma} =  L^1(\RR)$.
\end{prop}

\noindent {\bf Proof:}  The expression for $\zeta$ is given by the 
above corollary as  
$$
\frac{(1 - 2^{1-\sigma - it})}{\sigma + it}\zeta(\sigma + it) = 
\int_{0}^\infty e^{-it\lambda} f_\sigma(\lambda) d\lambda.
$$
Since $1/2 < \sigma < 1$, it is clear by inspection that the right hand side
integral vanishes  for any t if and only if $\sigma + it$ is a zero of
the function $\zeta$.  On the other hand for a fixed $\sigma$, the
right hand side of the above equation is just
$\widehat{f_\sigma}(t)$.  Wiener's theorem give a precise condition
for the Fourier transform of a function to be zero free, which when
applied yields the result. 


\begin{thm}\label{RH}
Let $f_\sigma$ and $X_{f_\sigma}$ be as in the above Proposition. 
Then the Riemann hypothesis is valid if and only if  for each 
$\sigma \in (1/2, 1)$, there is a function 
$g_\sigma$ with zero free Fourier transform such that $g_\sigma \in
X_{f_\sigma}$.
\end{thm}

\noindent {\bf Proof:} Suppose $\sigma$ is such that
 $\widehat{f_\sigma}(t_0) = 0 $, but  
there is a $g_\sigma \in L^1(\RR)$ such that
$\widehat{g_\sigma}(t) \neq 0 $ for any $t\in \RR$ but $g_\sigma \in
X_{f_\sigma}$.  Then since $g_\sigma \in X_\sigma$, we can find for any
$\epsilon > 0$, complex numbers $c_1, \cdots, c_{n(\epsilon)}$ and real numbers 
$x_1, \cdots, x_{n(\epsilon)}$ such that
$$
\|\sum_{i = 1}^n c_n f_\sigma(\cdot - x_i) - g_\sigma(\cdot )\|_1 <
\epsilon.
$$  
But 
\begin{align}
|\widehat{g_\sigma}(t_0)| &= |\widehat{g_\sigma}(t_0) - \sum_{i=1}^n c_n
e^{i t_0 x_i}\widehat{f_\sigma}(t_0) | \\
&\leq  
\|\sum_{i = 1}^n c_n f_\sigma(\cdot - x_i) - g_\sigma(\cdot )\|_1 <
\epsilon.
\end{align}
The epsilon being arbitrary, it follows that $g_\sigma(t_0) = 0$,
giving a contradiction.  Therefore $\widehat{f_\sigma}$ is zero free
under the assumption of the theorem.  Now the
equivalence follows from Wiener's Tauberian theorem.  

Since the Fourier transform of the convolution of two $L^1$ functions 
is the product of their Fourier transforms it is obvious that in the
above theorem we could replace $f_\sigma$ by its convolution with any
integrable function of zero free Fourier transform.  This we state
as  a corollary.  

\begin{cor}\label{RH1}
Let $f_\sigma$ and $X_{f_\sigma}$  be defined as in the above theorem.
Then the Riemann hypothesis is valid if and only if  for each 
$\sigma \in (1/2, 1)$, there is a pair of functions $h_\sigma, g_\sigma$
(not necessarily distinct), with zero free Fourier transforms such
that $g_\sigma \in X_{f_\sigma* h_\sigma}$.
\end{cor}

{\bf Acknowledgement:} We thank  R Balasubramanian and  V S
Sunder for discussions that helped us in this work.
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\end{document}
