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, $\Xi (t)=\xi \left(\frac{1}{2}+it\right)$. 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 $s=\sigma +it=\frac{1}{2}+\lambda +it$. We then show that the function $S_1(\omega )$ studied by Pólya 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.

15 pages