Intriguingly, various constructions related to Brownian motion result in quantities with moments described by the Riemann zeta function. These distributions appear in integral representations used to extend the zeta function to the entire complex plane, as described in an earlier post. Now, I look at how they also arise from processes constructed from Brownian motion such as Brownian bridges, excursions and meanders.

Recall the definition of the Riemann zeta function as an infinite series

which converges for complex argument *s* with real part greater than one. This has a unique extension to an analytic function on the complex plane outside of a simple pole at *s* = 1.

Often, it is more convenient to use the Riemann xi function which can be defined as zeta multiplied by a prefactor involving the gamma function,

This is an entire function on the complex plane satisfying the functional equation *ξ*(1 - *s*) = *ξ*(*s*).

It turns out that *ξ* describes the moments of a probability distribution, according to which a random variable *X* is positive with moments

(1) |

which is well-defined for all complex *s*. In the post titled The Riemann Zeta Function and Probability Distributions, I denoted this distribution by Ψ, which is a little arbitrary but was the symbol used for its probability density. A related distribution on the positive reals, which we will denote by Φ, is given by the moments

(2) |

which, again, is defined for all complex *s*.

As standard, complex powers of a positive real *x* are defined by *x*^{s} = *e*^{slogx}, so (1,2) are equivalent to the moment generating functions of log*X*, which uniquely determines the distributions. The probability densities and cumulative distribution functions can be given, although I will not do that here since they are already explicitly written out in the earlier post. I will write *X* ∼ Φ or *X* ∼ Ψ to mean that random variable *X* has the respective distribution. As we previously explained, these are closely connected:

- If
*X*∼ Ψ and, independently,*Y*is uniform on [1, 2], then*X*/*Y*∼ Φ. - If
*X*,*Y*∼ Φ are independent then √*X*^{2}+*Y*^{2}∼ Ψ.

The purpose of this post is to describe some constructions involving Brownian bridges, excursions and meanders which naturally involve the Φ and Ψ distributions.

Theorem 1The following have distributionΦ:

- √2/π
ZwhereZ= sup_{t}|B_{t}|is the absolute maximum of a standard Brownian bridgeB.Z/√2πwhereZ= sup_{t}B_{t}is the maximum of a Brownian meanderB.- √2π
ZwhereZis the sample standard deviation of a Brownian bridgeB,

with sample meanB̅= ∫_{0}^{1}B_{t}dt.- √π/2
ZwhereZis the pathwise Euclidean norm of a 2-dimensional Brownian bridgeB= (B^{1},B^{2}),

- √
τπ/2whereτ= inf{t≥ 0: ‖B_{t}‖= 1}is the first time at which the norm of a 3-dimensional standard Brownian motionB= (B^{1},B^{2},B^{3})hits1.

The Kolmogorov distribution is, by definition, the absolute maximum of a Brownian bridge. So, the first statement of theorem 1 is saying that Φ is just the Kolmogorov distribution scaled by the constant factor √2/π. Moving on to Ψ;

Theorem 2The following have distributionΨ:

- √2/π
ZwhereZ= sup_{t}B_{t}– inf_{t}B_{t}is the range of a standard Brownian bridgeB.- √2/π
ZwhereZ= sup_{t}B_{t}is the maximum of a (normalized) Brownian excursionB.- √π/2
ZwhereZis the pathwise Euclidean norm of a 4-dimensional Brownian bridgeB= (B^{1},B^{2},B^{3},B^{4}),

Continue reading “Brownian Motion and the Riemann Zeta Function”