Some results on the $\xi(s)$ and $\Xi(t)$ functions associated with Riemann's $\zeta(s)$ function

Abstract

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.