Some results on the $ξ(s)$ and $Ξ(t)$ functions associated with Riemann's $ζ(s)$ function

Abstract

We report on some properties of the $ξ(s)$ function and its value on the critical line, $Ξ(t)=ξ\left(\tfrac{1}{2}+it\right)$. 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 $ξ(s)$ function, and present a simple proof of the horizontal monotonicity of the modulus of $ξ(s)$. We then show that the $Ξ(t)$ function can be interpreted as the autocorrelation function of a weakly stationary random process, whose power spectral function $S(ω)$ and $Ξ(t)$ form a Fourier transform pair. We then show that $ξ(s)$ can be formally written as the Fourier transform of $S(ω)$ into the complex domain $τ=t-iλ$, where $s=σ+it=\tfrac{1}{2}+λ+it$. We then show that the function $S_1(ω)$ studied by Pólya has $g(s)$ as its Fourier transform, where $ξ(s)=g(s)ζ(s)$. 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.