Newton flow and number theory

A central problem of analytic number theory is the Riemann Hypothesis, describing a conjecture about the zeros of the Riemann zeta-function \(\zeta(s)=\sum_{n=1}^{\infty} n^{-s},\) where \(s\equiv\sigma + i \tau\) is the complex argument. The conjecture says that all non-trivial zeros of \(\zeta\) lie on a critical axis with \(\sigma=1/2.\)

The same conjecture can be formulated for the Riemann \(\xi\)-function, $$\xi(s)\equiv\pi^{-s/2}(s-1)\Gamma(s/2-1)\zeta(s),$$ since it has the same non-trivial zeros as \(\zeta.\) Here \(\Gamma\) is the Gamma function. Moreover, \(\xi \)-function has no poles and leads to the functional equation \(\xi(s)=\xi(1-s).\)

There have been numerous investigations of these functions so far and most of them rely heavily on analytic estimates. Relatively few geometric considerations have been pursued.

We study properties of the Riemann \(\xi\)-function from a geometrical point of view using the Newton flow method defined by $$\dot{s}(t)=-\frac{\xi(s)}{\xi'(s)},$$ and leading us to lines of constant phase of \(\xi\). Since \(\xi\) has no poles, the lines of constant phase start at infinity and terminate at a zero of the function. This approach effectively divides the complex plane into individual regions by special phase lines called separatrices. Every such region serves as the support of the incoming flow into one specific zero of \(\xi.\) This geometrical approach promises to be a powerful tool for studies of the zeros of \(\xi\) and \(\zeta\).