Brownian motion and Riemann zeta function

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

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

satisfying the functional equation

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

It turns out that $2\backslash xi(s)$ describes the moments of a probability distribution $\backslash mu${{cite book |author=Roger Mansuy and Marc Yor |title=Aspects of Brownian Motion |series=Universitext |publisher=Springer, Berlin, Heidelberg |year=2008 |pages=165–167 |language=en |doi=10.1007/978-3-540-49966-4|isbn=978-3-540-22347-4 }}

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

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 $b\_s$ describes the same distribution. A Brownian bridge is a one-dimensional Brownian motion $(W\_t)\_\{0\backslash leq\; t\backslash leq\; 1\}$ conditioned on $W\_0=W\_1=0$.{{cite journal |author=Philippe Biane and Marc Yor |title=Valeurs principales associées aux temps locaux browniens |journal=Bulletin de Science Mathématique |volume=111 |year=1987 |pages=23–101 |language=fr}} They showed that

:$X=\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\backslash left(\backslash max\backslash limits\_\{0\backslash leq\; s\; \backslash leq\; 1\}\; b\_s-\backslash min\backslash limits\_\{0\backslash leq\; s\; \backslash leq\; 1\}\; b\_s\backslash right)$

is a solution for the moment equation

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

However, this is not the only process that follows this distribution.

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

:$R\_t:=\backslash sqrt\{\backslash left(W\_t^\{(1)\}\backslash right)^2+\backslash left(W\_t^\{(2)\}\backslash right)^2+\backslash left(W\_t^\{(3)\}\backslash right)^2\}.$

Define the hitting time $T\_1:=\backslash inf\backslash \{t\backslash geq\; 0\backslash colon\; R\_t=1\backslash \}$ and let $\backslash tilde\{T\_1\}$ be an independent hitting time of another $\backslash operatorname\{Bes\}(3)$ process. Define the random variable

:$N=\backslash frac\{\backslash pi\}\{2\}\backslash left(T\_1+\backslash tilde\{T\_1\}\backslash right),$

then we have

:$2\backslash xi(2s)=\backslash mathbb\{E\}[N^s].${{cite journal |author=Philippe Biane, Jim Pitman, and Marc Yor |title=Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions |journal=Bulletin of the American Mathematical Society |volume=38 |year=2001 |issue=4 |pages=435–465 |doi=10.1090/S0273-0979-01-00912-0 |arxiv=math/9912170}}

Let $\backslash varphi$ be the Radon–Nikodym density of the distribution $\backslash mu$, then the density satisfies the equation{{cite book |author=Roger Mansuy and Marc Yor |title=Aspects of Brownian Motion |series=Universitext |publisher=Springer, Berlin, Heidelberg |year=2008 |pages=165–167 |language=en |doi=10.1007/978-3-540-49966-4|isbn=978-3-540-22347-4 }}

:$\backslash varphi(t):=2t\; \backslash Theta\text{'}\text{'}(t)\; +\; 3\backslash Theta\text{'}(t)$

for the theta function

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

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

:$h(y)=2y\; G\text{'}(y)+y^2G\text{'}\text{'}(y).$

with explicit form

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

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

:$2\backslash xi(s)=\backslash int\_0^\{\backslash infty\}\; y^\{s-1\}h(y)dy.$

• {{cite book

|author=Roger Mansuy and Marc Yor

|title=Aspects of Brownian Motion

|series=Universitext

|publisher=Springer, Berlin, Heidelberg

|year=2008

|language=en

|doi=10.1007/978-3-540-49966-4|isbn=978-3-540-22347-4

}}

Category:Probability theory