Brownian motion and Riemann zeta function

In mathematics, the Brownian motion and the Riemann zeta function are two central objects of study in mathematics originating from different fields - probability theory and analytic number theory - that have mathematical connections between them. The relationships between stochastic processes derived from the Brownian motion and the Riemann zeta function show in a sense intuitively the stochastic behaviour underlying the Riemann zeta function. A representation of the Riemann zeta function in terms of stochastic processes is called a stochastic representation.

Brownian Motion and the Riemann Zeta Function

Let denote the Riemann zeta function and the gamma function, then the Riemann xi function is defined as
satisfying the functional equation
It turns out that describes the moments of a probability distribution

Brownian Bridge and Riemann Zeta Function

In 1987 Marc Yor and Philippe Biane proved that the random variable defined as the difference between the maximum and minimum of a Brownian bridge describes the same distribution. A Brownian bridge is a one-dimensional Brownian motion conditioned on. They showed that
is a solution for the moment equation
However, this is not the only process related to this distribution, for example the Bessel process also gives rise to random variables with the same distribution.

Bessel process and Riemann Zeta Function

A Bessel process of order is the Euclidean norm of a -dimensional Brownian motion. The process is defined as
where is a -dimensional Brownian motion.
Define the hitting time and let be an independent hitting time of another process. Define the random variable
then we have

Distribution

Let be the Radon–Nikodym density of the distribution, then the density satisfies the equation
for the theta function
An alternative parametrization yields
with explicit form
where and