Chaos expansion of uniformly distributed random variables and application to number theory

Abstract

The chaos expansion of a random variable with uniform distribution is given. This decomposition is applied to analyze the behavior of each chaos component of the random variable $\log \zeta $ on the so-called critical line, where ζ is the Riemann zeta function. This analysis gives a better understanding of a famous theorem by Selberg.

Keywords

• Riemann zeta function
• Malliavin calculus
• multiple Wiener–Itô integrals
• Selberg theorem