The Riemann zeta function on the critical line

Dicembre 6th, 2022 | by Marcello Colozzo |

The graphic animation in the figure represents the motion of two balls whose hourly equation is composed of the real part and the imaginary part of the Riemann zeta function calculated on the critical line, changed sign and normalized on its argument.

Let ζ(z) denote the Riemann zeta function. As known

where

being [.] the integer part function. It follows that

is the Fourier transform of u(t). To study the behavior of this real function, let us first focus on [e^t]. Trivially

so we find the trend shown in Fig.: the graph is made up of the union of the representative half-line of the unlimited interval (-oo,0) and of a countable infinity of segments of decreasing and infinitesimal length to infinity.

It is easy to understand the trend of the graph of the function e^{-t/2}[e^{t}] which we report in fig.

from which we see that the function is divergent for t->+oo. The graph of u(t) is shown in fig.

We therefore note that u(t) is a sawtooth function, where the height of the teeth decreases exponentially.

In fig.

we report the trend of u(t) for t variable from -10 to +10, and this gives the idea of the behavior at infinity. Precisely, the function vanishes exponentially and this guarantees the convergence of the Fourier integral. In fig.