We investigate Riemann's xi function $\xi(s):=\frac{1}{2}s(s-1)\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ (here $\zeta(s)$ is the Riemann zeta function). The Hypothesis (RH) asserts that if $\xi(s)=0$, then $\mathrm{Re}(s)=\frac{1}{2}$. P\'olya proved RH equivalent to hyperbolicity of Jensen polynomials $J^{d,n}(X)$ constructed from certain Taylor coefficients $\xi(s)$. For each $d\geq 1$, ...

Introduction. Since the time of Riemann, there has been a close interplay between the study of the geometry of complex algebraic curves (or, equivalently, compact Riemann surfaces) and the topology of 2-manifolds. These connections arise from the uniformization theorem, which asserts that every simply connected Riemann surface is conformally equivalent to either the Riemann sphere, the plane, o...

In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function Ξ(t) = ξ( 1 2 + it). First, we prove that positivity of all local maxima and negativity of all local minima of Ξ(t) form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of Ξ(t) is a ...