Modern Stochastics: Theory and Applications

Let H be a real and separable Hilbert space and let $(W(h),h\in H)$ be an isonormal Gaussian process on a standard probability space $\left(\Omega ,\mathcal{F},P\right)$. It is well-known that any square integrable random variable, measurable with respect to the sigma-algebra generated by W can be decomposed in chaos, i.e. it can be written as an orthogonal sum $F={\textstyle\sum _{n\ge 0}}{F_{n}}$ where for every $n\ge 0$, the random variable ${F_{n}}$ is an element of the nth Wiener chaos. The knowledge of the concrete chaos expansion of F (i.e. the knowledge the exact form of ${F_{n}}$ for every $n\ge 0$) gives an important information on the random variable F. If F is a random variable with a given common probability distribution, it is in general hard to get its exact chaos expansion, except in some particular case (Gaussian distribution, Gamma distribution, etc). In this work, our purpose is to find the chaos expansion of a random variable with uniform distribution and to apply this result to a well-known problem in number theory. We will consider the random variable F given by with $f,g$ orthonormal elements of the Hilbert space H. Then F follows the uniform distribution over the unit interval $[0,1]$. By using the tools of Malliavin calculus, Stroock formula and the properties of the Hermite polynomials, we derive the chaos decomposition of the random variable defined by (1). Then we will apply our result to the Selberg theorem, which concerns the behavior of the Riemann zeta fumction on the critical line. Let us recall the context. The Riemann zeta function ζ is defined, for $\mathrm{\Re }s>1$, by and when $\mathrm{\Re }s\le 1$, the function ζ is defined as an analytic continuation of (2). The distribution of the zeta zeros is one of the outstanding problems in mathematics. It is known that $\zeta (-2n)=0$ for every $n\ge 1$. The points $-2n$ are called the trivial zeros of the Riemann zeta function. The most mysterious facts about the Riemann zeta function concerns the distribution of its nontrivial zeros. The Riemann hypothesis claims that all the nontrivial zeros of the Riemann zeta function lie on the critical line $\mathrm{\Re }s=\frac{1}{2}$. Therefore, the behavior of the function ζ on the critical line and close to this critical line has been intensively studied.

A famous result by Atle Selberg says that, if $T>0$ and t is a random variable uniformly distributed over the interval $[T,2T]$, then the sequence where ${X_{1}}+i{X_{2}}$ is a complex-valued standard normal random variable, i.e. ${X_{1}},{X_{2}}\sim N(0,1)$ are independent random variables. There are several versions of this theorem. In particular, the result (3) holds if $t\sim \mathcal{U}[0,T]$ or, more generally, if $t\sim \mathcal{U}[aT,bT]$ with $b>a\ge 0$. We will work with $t\sim \mathcal{U}[0,T]$ and we will assume throughout in the sequel $t=TU$ with $U\sim \mathcal{U}[0,1]$. As in the literature (e.g., [7, 16]), we will still call Selberg’s theorem the result concerning the convergence of (3) with $t=TU$.

Recall that for any $z\in \mathbb{C}$, $\log z=\log |z|+i\arg z$. Then the convergence (3) is actually equivalent to (4) and (5) below (see [11–13]) and where $”{\to ^{(d)}}”$ stands for the convergence in distribution.

The idea behind the proof of the limit theorems (4) and (5) is (see, among others, [11–15]) to approximate $\log \zeta \left(\frac{1}{2}+\mathbf{i}t\right)$ by the (renormalized) Dirichlet series with $t=TU$, $U\sim \mathcal{U}[0,1]$ and ε small enough. We will work throughout with $\varepsilon =1$. This approximation of $\log \zeta \left(\frac{1}{2}+\mathbf{i}t\right)$ by the Dirichlet series (6) is in ${L^{2}}$ sense, since (see [13], see also [7] for a detailed proof) where C is a strictly positive constant not depending on T. This implies that the sequence

Our purpose is to bring a new contribution to understanding the limit theorems (4) and (5). More exactly, we will analyze the asymptotic behavior of each chaos component of $\frac{1}{\sqrt{\frac{1}{2}\log \log T}}\log |\zeta \left(\frac{1}{2}+it\right)|$ with $t\sim \mathcal{U}[0,T]$. That is, if ${J_{n}}(T)$ denotes the projection of $\frac{1}{\sqrt{\frac{1}{2}\log \log T}}\log \left|\zeta \left(\frac{1}{2}+it\right)\right|$ on the nth Wiener chaos, we want to study the limit behavior in distribution of ${J_{n}}(T)$ as $T\to \infty $ for each fixed $n\ge 0$. This will allow to understand which chaos projection ${J_{n}}(T)$ is dominant with respect to the others and determines the limit behavior of the $\log \zeta $.

Many other related works in the old and recent literature treated the distribution of the zeros of the Riemann zeta function. Some related results, among many others, are [1–4, 6].

Our analysis will be based on the study of the Dirichlet approximation (6). Using the chaos expansion of the uniformly distributed random variable U, we will find the chaos expansion of the random variable $\cos (TU\log p)$ with $U\sim \mathcal{U}[0,1]$ via Malliavin calculus and we will study the limit in distribution of each chaos component. We will see that every chaos converges to zero, but their sum tends in distribution to the Gaussian law. All the chaoses contribute to the limit and there is no term that is bigger than the others and gives the limit behavior.

Our work has the following structure. Section 2 contains some preliminaries on Wiener chaos and Malliavin calculus needed throughout the work. Section 3 is devoted to the chaos expansion of a uniformly distributed random variable U. In Section 4, we obtain the chaos decomposition of the Dirichlet series that approximates $\log \zeta \left(\frac{1}{2}+\mathbf{i}TU\right)$ and we then analyze the asymptotic behavior, as $T\to \infty $, of each chaos component.

We also present the elements from the Malliavin calculus that will be used in the paper. We refer to [9] for a more complete exposition. Consider $\mathcal{H}$ as a real separable Hilbert space and $(B(\varphi ),\varphi \in \mathcal{H})$ an isonormal Gaussian process on a probability space $(\Omega ,\mathcal{A},\mathbb{P})$, that is, a centered Gaussian family of random variables such that $\mathbf{E}\left(B(\varphi )B(\psi )\right)={\langle \varphi ,\psi \rangle _{\mathcal{H}}}$.

We denote by D the Malliavin derivative operator that acts on smooth functions of the form $F=g(B({\varphi _{1}}),\dots ,B({\varphi _{n}}))$ (g is a smooth function with compact support and ${\varphi _{i}}\in \mathcal{H}$, $i=1,\dots ,n$)

It can be checked that the operator D is closable from $\mathcal{S}$ (the space of smooth functionals as above) into ${L^{2}}(\Omega ;\mathcal{H})$ and it can be extended to the space ${\mathbb{D}^{1,p}}$ which is the closure of $\mathcal{S}$ with respect to the norm

We denote by ${\mathbb{D}^{k,\infty }}:={\cap _{p\ge }}{\mathbb{D}^{k,p}}$ for every $k\ge 1$. In this paper, $\mathcal{H}$ will be the standard Hilbert space ${L^{2}}([0,T])$.

We will make use of the chain rule for the Malliavin derivative (see Proposition 1.2.4 in [9]). That is, if $\varphi :\mathbb{R}\to \mathbb{R}$ is a continuously differentiable function and $F\in {\mathbb{D}^{1,2}}$, then $\varphi (F)\in {\mathbb{D}^{1,2}}$ and

Denote by ${I_{n}}$ the multiple stochastic integral with respect to B (see [9]). This ${I_{n}}$ is actually an isometry between the Hilbert space ${H^{\odot n}}$ (symmetric tensor product) equipped with the scaled norm $\sqrt{n!}\| \cdot {\| _{{H^{\otimes n}}}}$ and the Wiener chaos of order n which is defined as the closed linear span of the random variables ${H_{n}}(B(\varphi ))$ where $\varphi \in \mathcal{H}$, $\| \varphi {\| _{H}}=1$ and ${H_{n}}$ is the Hermite polynomial of degree $n\ge 1$ The isometry of multiple integrals can be written as: for $m,n$ positive integers, The Malliavin derivative D acts on the Wiener chaos as an annilihilation operator: if $F={I_{n}}(f)$ with symmetric $f\in {L^{2}}({[0,T]^{n}})$, then ${D_{t}}F=n{I_{n-1}}(f(\cdot ,t))$ where “·” stands for $n-1$ variables in $[0,T]$.

The uniformly distributed random variable U will be chosen of a particular form that allows to use the techniques of the Malliavin calculus. Actually, in the sequel we will assume with the following conditions fixed throughout our work: $f,g\in H$, $\| f\| =\| g\| =1$ and $\langle f,g\rangle =0$ (all the scalar products and norms in the paper will be considered in H if no further specification is made). In (10), $(W(h),h\in H={L^{2}}([0,1]))$ stands for a Gaussian isonormal process as described in Section 2. In particular, this implies that $W(f)$ and $W(g)$ are independent standard normal random variables. The fact that the random variable (10) is uniformly distributed over $[0,1]$ follows from the simple computations: with $F:\mathbb{R}\to \mathbb{R}$ an arbitrary function such that $\mathbf{E}F(U)<\infty $, where we used the change of variables with polar coordinates.

We analyze the chaos expansion of for every $k>0$. This will be done by using the techniques of the Malliavin calculus. We will also need some properties of the Hermite polynomials. Let ${H_{n}}$ denote the nth Hermite polynomial, see (8). Recall that, if $Y\sim N(0,{\sigma ^{2}})$, then and $\mathbf{E}{H_{n}}(Y)=0$ if n is odd.

We will use the following two auxiliary lemmas that concern the Hermite polynomials.

Recall that ${D^{(s)}}(s\ge 1)$ denotes the sth iterated Malliavin derivative. If $s=0$ then by convention ${f^{\otimes s}}=1$.

The next step is to get the chaos expansion of ${U^{2k}}$ with U a random variable uniformly distributed over $[0,1]$. If $f,g$ are two functions, by $f\tilde{\otimes }g$ we denote the symmetrization of their tensor product.

We use the results in the previous section in order to get the chaos expansion of the Dirichlet series (6) and to study the behavior, as $T\to \infty $, of each chaos component of $\log \zeta \left(\frac{1}{2}+\mathbf{i}TU\right)$ with U given by (10).