Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some results on the $\xi(s)$ and $\Xi(t)$ functions associated with
  Riemann's $\zeta(s)$ function   We report on some properties of the $\xi(s)$ function and its value on the
critical line, $\Xi(t)=\xi\left(\tfrac{1}{2}+it\right)$. First, we present some
identities that hold for the log derivatives of a holomorphic function. We then
re-examine Hadamard's product-form representation of the $\xi(s)$ function, and
present a simple proof of the horizontal monotonicity of the modulus of
$\xi(s)$. We then show that the $\Xi(t)$ function can be interpreted as the
autocorrelation function of a weakly stationary random process, whose power
spectral function $S(\omega)$ and $\Xi(t)$ form a Fourier transform pair. We
then show that $\xi(s)$ can be formally written as the Fourier transform of
$S(\omega)$ into the complex domain $\tau=t-i\lambda$, where
$s=\sigma+it=\tfrac{1}{2}+\lambda+it$. We then show that the function
$S_1(\omega)$ studied by P\'{o}lya has $g(s)$ as its Fourier transform, where
$\xi(s)=g(s)\zeta(s)$. Finally we discuss the properties of the function
$g(s)$, including its relationships to Riemann-Siegel's $\vartheta(t)$
function, Hardy's Z-function, Gram's law and the Riemann-Siegel asymptotic
formula.

17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc