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 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.^[1][2]

Let ${\displaystyle \zeta (s)}$ denote the Riemann zeta function and ${\displaystyle \Gamma }$ the gamma function, then the Riemann xi function is defined as

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

satisfying the functional equation

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

It turns out that ${\displaystyle 2\xi (s)}$ describes the moments of a probability distribution ${\displaystyle \mu }$^[2][3]

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

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 ${\displaystyle (b_{s})_{0\leq s\leq 1}}$ describes the same distribution. A Brownian bridge is a one-dimensional Brownian motion ${\displaystyle (W_{t})_{0\leq t\leq 1}}$ conditioned on ${\displaystyle W_{0}=W_{1}=0}$.^[4] They showed that

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

is a solution for the moment equation

${\displaystyle 2\xi (s)=\mathbb {E} [X^{s}]}$

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.

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

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

where ${\displaystyle W_{t}=(W_{t}^{(1)},W_{t}^{(2)},W_{t}^{(3)})}$ is a ${\displaystyle 3}$-dimensional Brownian motion.

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

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

then we have

${\displaystyle 2\xi (2s)=\mathbb {E} [N^{s}].}$^[5][2]

Let ${\displaystyle \varphi }$ be the Radon–Nikodym density of the distribution ${\displaystyle \mu }$, then the density satisfies the equation^[2]

${\displaystyle \varphi (t)=2t\Theta ''(t)+3\Theta '(t)}$

for the theta function^[2]

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

An alternative parametrization ${\displaystyle G(y):=\Theta (y^{2})}$ yields^[5]

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

with explicit form

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

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

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

• Montgomery's pair correlation conjecture