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]

Brownian Motion and the Riemann Zeta Function

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]

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

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 ${\displaystyle (b_{s})_{0\leq s\leq 1}}$  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.

Bessel process and Riemann Zeta Function

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]

Distribution

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

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

See also

• Montgomery's pair correlation conjecture

References

1. David Williams (1990). "Brownian motion and the Riemann zeta-function" (PDF). Disorder in Physical Systems: A Volume in Honour of John M. Hammersley. New York: Oxford University Press. pp. 361–372. ISBN 0198532156.
2. Roger Mansuy; Marc Yor (2008). Aspects of Brownian Motion. Universitext. Berlin, Heidelberg: Springer. pp. 165–167. doi:10.1007/978-3-540-49966-4. ISBN 978-3-540-22347-4.
3. Laurel Smith; Persi Diaconis (1988). "Honest bernoulli excursions". Journal of Applied Probability. 25 (3): 464–477. doi:10.2307/3213976. JSTOR 3213976.
4. Philippe Biane; Marc Yor (1987). "Valeurs principales associées aux temps locaux browniens". Bulletin des Sciences Mathématiques (in French). 111: 23–101.
5. Philippe Biane; Jim Pitman; Marc Yor (2001). "Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions". Bulletin of the American Mathematical Society. New Series. 38 (4): 435–465. arXiv:math/9912170. doi:10.1090/S0273-0979-01-00912-0.