Brownian motion and Riemann zeta function

In mathematics, the Brownian motion and the Riemann zeta function are two central objects of study originating from different fields - probability theory and analytic number theory - that have deep mathematical connections between them. The relationships between stochastic processes derived from the Brownian motion and the Riemann zeta function show in a sense inuitively 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 $\zeta (s)$ denote the Riemann zeta function and $\Gamma$ the gamma function, then the Riemann xi function is defined as

$\xi (s):={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma ({\tfrac {1}{2}}s)\zeta (s)$

satisfying the functional equation

$\xi (s)=\xi (1-s),\quad \forall s\in \mathbb {C} .$

It turns out that $2\xi (s)$ describes the moments of a probability distribution $\mu$

$2\xi (s)=\mathbb {E} [X^{s}]=\int _{0}^{\infty }x^{s}d\mu (x),\quad \forall s\in \mathbb {C}$

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 $b_{s}$ describes the same distribution. A Brownian bridge is a one-dimensional Brownian motion $(W_{t})_{0\leq t\leq 1}$ conditioned on $W_{0}=W_{1}=0$. They showed that

$X={\sqrt {\frac {2}{\pi }}}\left(\max \limits _{0\leq s\leq 1}b_{s}-\min \limits _{0\leq s\leq 1}b_{s}\right)$

is a solution for the above moment equation. However, this is not the only process that follows this distribution.

Bessel Process and Riemann Zeta Function

A Bessel process $\operatorname {Bes} (d)$ of order $d$ is the Euclidean norm of a $d$-dimensional Brownian motion. The $\operatorname {Bes} (3)$ process is defined as

$R_{t}:={\sqrt {\left(W_{t}^{(1)}\right)^{2}+\left(W_{t}^{(2)}\right)^{2}+\left(W_{t}^{(3)}\right)^{2}}}.$

Define the hitting time $T_{1}:=\inf\{t\geq 0\colon R_{t}=1\}$ and let ${\tilde {T_{1}}}$ be an independent hitting time of another $\operatorname {Bes} (3)$ process. Define the random variable

$N={\frac {\pi }{2}}\left(T_{1}+{\tilde {T_{1}}}\right),$

then we have

$2\xi (2s)=\mathbb {E} [N^{s}].$

Distribution

Let $\varphi$ be the Radon–Nikodym density of the distribution $\mu$, then the density satisfies the equation

$\varphi (t):=2t\Theta ''(t)+3\Theta '(t)$

for the theta function

$\Theta (t)=\sum \limits _{n=-\infty }^{\infty }e^{-\pi n^{2}t}.$

An alternative parametrization $G(y):=\Theta (y^{2})$ yields

$h(y)=2yG'(y)+y^{2}G''(y).$

with explicit form

$h(y)=4y^{2}\sum \limits _{n=1}^{\infty }\pi (2\pi n^{4}y^{2}-3n^{2})e^{-\pi n^{2}y^{2}}$

where $h(y)=2y\varphi (y^{2})$ and

$2\xi (s)=\int _{0}^{\infty }y^{s-1}h(y)dy.$

Bibliography

• Roger Mansuy and Marc Yor (2008). Aspects of Brownian Motion. Universitext. Springer, Berlin, Heidelberg. doi:10.1007/978-3-540-49966-4. ISBN 978-3-540-22347-4.