We establish a Brownian extension to Selberg’s central limit theorem for the Riemann zeta function. This implies various limiting distributions for \(\zeta\), including an analogue of the reflection principle for the maximum of the Brownian motion: as \(T\) diverges, for any \(u>0\) we have \[\frac{1}{T}\cdot {\rm meas}\Big\{0\leqslant t\leqslant T:\max_{\sigma\geqslant\tfrac{1}{2}}\log|\zeta(\sigma+i t)|\geqslant u \sqrt{\tfrac{1}{2}\log \log T} \Big\}\to 2\int_u^{\infty} \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi}}{\rm d}x.\]

section10mm-21Introduction

subsection20mm-0 ****Statistics on \(\zeta\). Selberg’s central limit theorem [@Sel1946] states that for \(\tau\) is chosen uniformly at random from \([0, T]\),

\[\label{clt} \dfrac{1}{\sqrt{\log \log T}}\, \log \zeta\big(\frac{1}{2}+ i \tau\big) \overset{\text{law}}{\longrightarrow} \mathscr{N}_{\mathbb{C}}\tag{1}\] as \(T \rightarrow + \infty\), where \(\mathscr{N}_{\mathbb{C}}\) is a standard complex Gaussian random variable, i.e. with independent normal real and imaginary parts of variance \(\tfrac{1}{2}\). A simple proof for \(\log |\zeta|\) can be found in [@RadSou2017].

This theorem was the first major result on the statistical behaviour of \(\zeta\) around the critical line; since then, many advances in probabilistic number theory involve new statistics for \(\zeta\). A prominent example is Montgomery’s pair correlation [@pair], which conjecturally identifies the local spacings between high zeros of Riemann zeta function with the scaling limit of random matrices from the Gaussian unitary ensemble.

More recently, the Fyodorov-Hiary-Keating conjecture [@FyoHiaKea2012; @fyodorov2014freezing] proposed to extend this analogy to extreme values statistics, through precise estimates for the maximum of the Riemann zeta function over short intervals on the critical line. In particular, these papers conjectured that, if \(\tau \in [T, 2T]\) is chosen uniformly at random, \[\label{FHK} \max_{|h| \leqslant 1} |\zeta(\frac{1}{2}+ i \tau + i h)| \asymp \frac{\log T}{(\log \log T)^{\frac{3}{4}}}\tag{2}\] meaning that the ratio between the two sides is tight as \(T \to \infty\). After initial progress in [@Naj16; @ArgBelBouRadSou2016; @Har2019], these estimates on this ratio and its universal tail asymptotics were proved in [@arguin2020fyodorov1; @arguin2020fyodorov2].

One natural question we consider in this paper is about the maximum of \(\zeta\) over horizontal intervals. Our main result is to control the behaviour of \(\zeta\) in intervals of the form \([\frac{1}{2}+ i \tau, \frac{1}{2}+ \eta + i \tau]\) by exhibiting a form of functional tightness. This - for example - will allow us to show that for large \(T\) \[\label{eqn:maxPr} \frac{1}{\sqrt{\log \log T}}\max_{\sigma \geqslant\tfrac{1}{2}} \log|\zeta( \sigma + i \tau)|\overset{\text{law}}{\longrightarrow} |\mathscr{N}(0,\frac{1}{2})|,\tag{3}\] where, again, \(\tau \in [0,T]\) is chosen uniformly at random. In particular, while the maximum (2 ) over vertical short intervals exhibits a subtle, larger scaling than (1 ), the size order for the maximum over horizontal intervals agrees with the normalization in Selberg’s central limit theorem, but the limiting random variable is the absolute value of a Gaussian instead of a Gaussian, mirroring the reflection principle for the Brownian motion.

subsection20mm-0 ****Main result. Our main statement is as follows.

The above convergence means that for any \(F: \mathcal{C}^0 \to \mathbb{R}\) a bounded functional which is continuous for the \({\rm L}^\infty([0,1])\) norm, we have \[\mathbb{E}[F(Z^{(T)})] \underset{T \to + \infty}{\longrightarrow} \mathbb{E}[F(B)].\]

Theorem 1 implies that for any \(0 \leqslant\alpha_1, \ldots, \alpha_n \leqslant 1\) the joint limit distribution of \((Z^{(T)}(\alpha_i))_{1 \leqslant i \leqslant n}\) is that of a Gaussian vector (with an appropriate correlation structure, described in Theorem 3). However, this finite-dimensional limit is already well-understood: Our proof of Theorem 3 is only a repurposing of the methods from [@bourgade]. Our main contribution is instead the fact that the sequence of random functions \((Z^{(T)})_{T \geqslant 0}\) is tight, which is not obvious and is necessary to deduce corollaries such as (3 ).

Finally, we note that for random unitary matrices, there is an analogous result to the finite dimensional convergence, Theorem 3, also shown in [@bourgade]. One might wonder whether we can also generalise this to a functional convergence towards Brownian motion, similarly to Theorem 1. More specifically, if \(U_n\) is a random, Haar-distributed, \(n\times n\) unitary matrix,, the relevant analogue of \(Z^{(T)}\) to consider would be \[\mathcal{Z}^{(n)}: \left. \begin{array}{rcl} [0, 1] & \longrightarrow &\mathbb{C}\\ \alpha & \longmapsto & \dfrac{1}{\sqrt{\log n}}\log\det(\exp(n^{- \alpha})I_n - U_n).\\ \end{array} \right.\] It seems likely that this process converges in distribution to a standard, complex Brownian motion as \(n\to\infty\).

subsection20mm-0 ****Consequences. Convergence in law to Brownian motion gives a few corollaries regarding the global behaviour of the Riemann zeta function in the critical strip; to the author’s knowledge, these corollaries are novel. Similar statements hold if we replace \(\log |\zeta|\) with \(\mathop{\mathrm{Im}}\log \zeta\).

The next result gives the distribution of the logarithmic measure of \(\sigma\) such that \(\log|\zeta(\sigma+ i\tau)|\geqslant 0\).

A third consequence of our theorem is a result quantifying how abnormally large the maximum around \(\tfrac{1}{2}+ i\tau\) (approached horizontally) becomes.

This originates from the so-called iterated logarithm law for Brownian motion, which follows from a similar law for random walks [@itlog]. The usual statement is that, for a real Wiener process \((X_t)_{t \geqslant 0}\),

\[\limsup_{t \to + \infty} \frac{X_t}{\sqrt{2t \log \log t}} = 1 \text{ a.s.}\] but, noting that \((t \mapsto t X_{\frac{1}{t}})\) is also a Wiener process, an analogous law follows for \(t \to 0\).

Another corollary of our functional convergence is the limiting occupation measure for \(\log |\zeta|\), on horizontal lines. More precisely, for \(t > 0\), let \(L^{(T)}_t\) be the local time of \(\mathop{\mathrm{Re}}Z^{(T)}\) i.e. the (almost everywhere) unique function \(L_t : \mathbb{R}\rightarrow \mathbb{R}_+\) such that, for any function \(\varphi\in \mathcal{C}^0(\mathbb{R}, \mathbb{R})\), \[\int_{0}^t \varphi(\mathop{\mathrm{Re}}Z^{(T)}(u)) du = \int_{\mathbb{R}} \varphi(v) L_t(v) dv.\]

This result grants us insight into the distribution of values of \(\log \zeta\). For instance:

subsection20mm-0 ****Notations and acknowledgments We use the convention \(f \ll g\) to mean \(f = O(g)\); if the implied constant depends upon another variable \(\varepsilon\), we shall write \(f \ll_\varepsilon g\). \(x \land y\) denotes the minimum of \(x\) and \(y\), and \(x^+ = \max(x, 0)\) is the positive part of \(x\). \(\log \zeta\) is defined in the usual way (see [@Sel1946] for example).

The author wishes to thank Paul Bourgade for his many helpful comments.

section10mm-21Overview of the proof of Theorem 1 We must prove two points in order to establish convergence in distribution: (i) convergence of finite-dimensional distributions; (ii) tightness of our process \(Z^{(T)}\).

subsection20mm-0 ****Convergence of finite-dimensional distributions. In this section, we prove the following theorem, as a first step towards establishing Theorem 1. For complex variables, we define \(\mathop{\mathrm{Cov}}(X,Y)=\mathbb{E}[{\bar X}Y]-\mathbb{E}[{\bar X}]\mathbb{E}[Y]\)

In order to achieve this, we will make use of the following lemma from [@bourgade]:

In order to make use of this, we shall replace \(\log \zeta\) with a related Dirichlet series. Indeed, it is known that, if \(\sigma \geqslant\frac{1}{2}\),

\[\log \zeta (\sigma + i \tau) - \sum_{p \leqslant T} \frac{1}{p^{\sigma + i \tau}}\] is bounded in \(\mathcal{L}^2\) uniformly in \(T\), and thus converges in distribution to \(0\) once divided by \(\sqrt{\log \log T}\). Here, we shall take \[\sigma_i = \frac{1}{2}+ \frac{1}{(\log T)^{\alpha_i}} \text{ for }1 \leqslant i \leqslant n.\] Using the Cramér–Wold method, in order to show Theorem 3, it is thus sufficient to show that for all \(\mu_1, \ldots, \mu_n \in \mathbb{C}\), \[\frac{1}{\sqrt{\log \log T}}\sum_{l=1}^n \mu_l \log \zeta(\sigma_l + i \tau) \overset{\text{law}}{\longrightarrow} \mathscr{N}(0, a^2)\] \[\text{i.e. } \frac{1}{\sqrt{\log \log T}} \sum_{l = 1}^n \mu_l \sum_{p \leqslant T} \frac{1}{p^{\sigma_l + i \tau}} \overset{\text{law}}{\longrightarrow} \mathscr{N}(0, a^2)\] where \(a^2 = \displaystyle\sum_{1 \leqslant i, j \leqslant n} \mu_i \overline{\mu_j} (1 \land \alpha_i \land \alpha_j)\). Now, setting \[a_{p, T} = \dfrac{\mathbb{1}_{p \leqslant T}}{\sqrt{\log \log T}} \sum_{l = 1}^n \dfrac{\mu_l}{p^{\sigma_l}}\] we simply need to check that the prerequisites of Lemma 1 hold. \((i)\) is clearly true; \((iii)\) holds if we set \(m_T = T^{\frac{1}{\log \log T}}\). Finally, in order to check \((ii)\), we just need to show \[\sum_{p \leqslant T} \dfrac{1}{p^{\sigma_i + \sigma_j}} \sim (1 \land\alpha_i \land \alpha_j) \log \log T \text{ for any }1 \leqslant i, j \leqslant n.\] This is shown in Lemma 3.3 of [@bourgade], and we skip the proof here: it is similar to the proof of our Lemma 3 later on. All prerequisites having been checked, Theorem 3 is therefore proven.

subsection20mm-0 ****The tightness criterion. Before proving the tightness of \(Z^{(T)}\), let us state a criterion which will be crucial to the proof. It is a modified version of a statement by Prokhorov [@prokh], which itself is adapted from a criterion by Kolmogorov for the continuity of stochastic processes.

The original statement of this theorem does not include \(\mathbb{1}_{A_\varepsilon^T}\); this is a relatively minor change, but we nevertheless include a proof for completeness.

If we want to extend Theorem 1 to \(a \in [0, \infty)\) (as in Remark 2), the criterion above does not apply as-is. Instead, we will note that

\[\mathbb{E}[\sup_{a \geqslant 1} |Z^{(T)}(a) - Z^{(T)}(1)|^4 \mathbb{1}_{A_\varepsilon^T}] \ll_\varepsilon\frac{1}{(\log \log T)^2}\]

via a straightforward modification of (11 ). This automatically guarantees convergence in law to our process \((\tilde{B}_\alpha)\) defined in (?? ).

subsection20mm-0 ****Structure of the proof of tightness. Theorem 4 allows us to transform a tightness problem, which would require some pretty strong uniform controls on \(\log \zeta\), into a moments calculation for which we have much better tools. In our case, we shall take \(A = 4\) and \(B=2\), owing to the roughly \(\frac{1}{2}\)-Holderian behaviour of Brownian motion.

Are the prerequisites of Theorem 4 verified by the process \(Z^{(T)}\)? The first one clearly is (taking for instance \(x_0 = 0\)) but the second one is much less obvious, and the purpose of the rest of this paper will be to prove that it holds. Specifically, from now on, we will set \(\varepsilon> 0\), and \(0 \leqslant a < b \leqslant 1\): for large enough \(T > 0\), our aim is to construct an adequate \(A_\varepsilon^T\) (independent of \(a, b\)) such that \[\label{momenttoshow} \mathbb{E}[|Z^{(T)}(a) - Z^{(T)}(b)|^4 \mathbb{1}_{A_\varepsilon^T}] \leqslant C_\varepsilon|a - b|^2.\tag{5}\] We will also set \[\sigma_1 = \frac{1}{2}+ \frac{1}{(\log T)^a} \text{ and } \sigma_2 = \frac{1}{2}+ \dfrac{1}{(\log T)^b}\] so that (5 ) becomes \[\mathbb{E}[|\log \zeta(\sigma_1 + i \tau) - \log \zeta(\sigma_2 + i \tau)|^4 \mathbb{1}_{A_\varepsilon^T}] \leqslant C_\varepsilon(b - a)^2(\log \log T)^2.\] In order to show this, we will proceed by approximating \(\log \zeta\) by a well-chosen Dirichlet sum. This allows us to effectively compute moments by expanding out the sum. Specifically, we will make use of the following decomposition from Selberg’s original paper on the CLT [@Sel1946]: for \(x > 1\), we may write \[\dfrac{\zeta'}{\zeta}(s) = -\sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{s}} + e_x(s)\] where \[\Lambda_x(n) = \begin{cases} \Lambda(n) & \text{ if } n \leqslant x \\ \Lambda(n) \dfrac{\log^2 \frac{x^3}{n} - 2 \log^2 \frac{x^2}{n}}{2 \log^2 x} & \text{ if } x \leqslant n \leqslant x^2 \\ \Lambda(n) \dfrac{\log^2 \frac{x^3}{n}}{2 \log^2 x} & \text{ if } x^2 \leqslant n \leqslant x^3 \end{cases}.\]

Here, we will be taking \(x = T^{\frac{1}{20}}\), although all of the following results are valid for \(x = T^{c}\) with small enough \(c\). As a result, \[\label{termstobound} \begin{align} \mathbb{E}[|Z^{(T)}(a) - Z^{(T)}(b)|^4 \mathbb{1}_{A_\varepsilon^T}] & \leqslant\dfrac{32}{(\log \log T)^2}\mathbb{E}\left[\left|\sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{\sigma_1 + i \tau} \log n} - \sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{\sigma_2 + i \tau} \log n} \right|^4 \right] \\ & + \dfrac{32}{(\log \log T)^2} \mathbb{E}\left[\left|\int_{\sigma_2}^{\sigma_1}e_x(\sigma + i \tau) d \sigma \right|^4\mathbb{1}_{A_\varepsilon^T}\right]. \end{align}\tag{6}\] The first term can be bounded quite effectively; this will be done in Section 4. To bound the second term, we will rely upon methods developed by Selberg in [@Sel1946]; this will be done in Section 5. In particular, the specific choice of \(\Lambda_x\) was made in order to be able to apply said methods.

With both terms bounded, we will mostly have proven our main theorem. However, setting \(\sigma_c = \dfrac{1}{2} + \dfrac{40 \log \frac{1}{\varepsilon}}{\log T}\), it turns out that this method breaks down when \(\sigma_2 \leqslant\sigma_c\). This case is not too complicated, and is handled separately at the end of the paper.

section10mm-21Moments of Dirichlet sums

As announced, in this section we will show the following proposition.

In order to prove Proposition 5, we shall use the two following technical lemmas:

These lemmas are very similar to existing results in the literature, but those are not quite sufficient for our purposes due to the dependence on \(\sigma, \sigma'\).

Assuming these lemmas, we may set \[\varphi(n) = \frac{\Lambda_x(n)}{\log n}(n^{-(\sigma_1 - \frac{1}{2})} - n^{-(\sigma_2 - \frac{1}{2})})\] and, applying Lemma 2, \[\begin{align} \mathbb{E}\left[\left|\sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{\sigma_1 + i \tau} \log n} - \sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{\sigma_2 + i \tau} \log n} \right|^4 \right] & \ll \left( \sum_{p \leqslant x^3 \text{ prime}} (p^{-\sigma_1} - p^{-\sigma_2})^2 \right)^2 \\ \end{align}\] \[= \left( \sum_{p \leqslant x^3} \left(\frac{1}{p^{2\sigma_1}} - \frac{1}{p^{\sigma_1 + \sigma_2}}\right) - \left( \frac{1}{p^{\sigma_1 + \sigma_2}} - \frac{1}{p^{2 \sigma_2}} \right) \right)^2.\] Applying Lemma 3, this shows Proposition 5.

We now just have to prove our other technical result.

section10mm-21The contribution of zeta zeroes

As a reminder, we have set \(\sigma_c = \dfrac{1}{2}+ \dfrac{40 \log \frac{1}{\varepsilon}}{\log T}\). We still have two points to handle in order to apply Theorem 4: (i) bounding the second term in (6 ); (ii) handling the case \(\sigma_2 \leqslant\sigma_c\).

In the interests of legibility, we will define (and work with) \(\eta_1 = \sigma_1 - \dfrac{1}{2}\), \(\eta_2 = \sigma_2 - \dfrac{1}{2}\), \(\eta_c = \sigma_c - \dfrac{1}{2}\), etc.

subsection20mm-0 ****Bounding the error \(e_x\).

Recall that we set

\[e_x(s) = \frac{\zeta'}{\zeta}(s) + \sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^s}.\] We wish to show that \[\label{toshow} \mathbb{E}\left[\left|\int_{\sigma_2}^{\sigma_1}e_x(\sigma + i \tau) d \sigma \right|^4\mathbb{1}_{A_\varepsilon^T}\right] \ll_\varepsilon(b-a)^2(\log \log T)^2\tag{8}\] when \(\sigma_c \leqslant\sigma_2 \leqslant\sigma_1\). Our main tool for showing this will be the following identity from [@Sel1946]

Applying this lemma, if \(\sigma_{x, t} \leqslant\sigma_2 \leqslant\sigma_1\) and \(T^{\frac{1}{2}} \leqslant t \leqslant T\),

\[\begin{align} \left|\int_{\sigma_2}^{\sigma_1}e_x(\sigma + i t) d \sigma \right| & \ll \left( \left|\sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{ \sigma_{x, t} + it}} \right| + \log T \right) \min(\frac{x^{-\frac{\eta_1}{2}}}{\log x}, (\sigma_1 - \sigma_2) x^{-\frac{\eta_2}{2}}). \end{align}\]

This is a good start, but how do we go from the condition \(\sigma_2 \geqslant\sigma_{x, t}\) to \(\sigma_2 \geqslant\sigma_c\)? For this, we need to show that \(\sigma_{x, t}\) is usually smaller than \(\sigma_c\): we will then cut out the region where \(\sigma_{x, t} \geqslant\sigma_c\) by choosing \(A_\varepsilon^T\) adequately. This is the object of the following lemma.

This means that we can set \(A_\varepsilon^T = ( \tau \notin Y_\varepsilon, \tau > \sqrt{T})\) and apply Lemma 5 to tackle (8 ): \[\begin{align} \mathbb{E}\left[\left|\int_{\sigma_2}^{\sigma_1}e_x(\sigma + i \tau) d \sigma\right|^4\mathbb{1}_{A_\varepsilon^T}\right] & \ll \left( \mathbb{E}\left[ \left|\sum_{n \leqslant x^3} \frac{\Lambda_x(n)}{n^{\sigma_{x, \tau} + i\tau}} \right|^4 \right] + (\log T)^4\right) \min(\frac{x^{-2\eta_1}}{(\log x)^4}, (\eta_1 - \eta_2)^4 x^{-2\eta_2}). \end{align}\]

To conclude, we will use the following proposition, whose proof will be given shortly:

Assuming Proposition 6, we can consider two cases. First, if \(b - a \leqslant\dfrac{1}{\log \log T}\), then \(\dfrac{\eta_1}{\eta_2} \ll 1\) and, applying (7 ), \[\begin{gather} \mathbb{E}[\int_{\sigma_2}^{\sigma_1}|e_x(\sigma + i \tau)|^4 d \sigma\mathbb{1}_{A_\varepsilon^T}] \ll_\varepsilon(\log T)^4 \eta_1^4 (b-a)^4 (\log \log T)^4 x^{-2\eta_2} \\ \ll (b-a)^4 (\eta_2 \log T)^4 e^{-\frac{1}{10} \eta_2 \log T}(\log \log T)^4 \ll (b-a)^2 (\log \log T)^2. \end{gather}\] If \(b-a \geqslant\dfrac{1}{\log \log T}\), then \[\mathbb{E}[\int_{\sigma_2}^{\sigma_1}|e_x(\sigma + i \tau)|^4 d \sigma\mathbb{1}_{A_\varepsilon^T}] \ll_\varepsilon\left(\frac{\log T}{\log x}\right)^4 x^{-2 \eta_2} \ll 1 \ll (b-a)^2 (\log \log T)^2.\] We therefore just need to show Proposition 6.

subsection20mm-0 ****Towards the critical line. As mentioned earlier, the case \(\sigma_2 \leqslant\sigma_c\) needs to be handled separately. Specifically, it still remains to be shown that:

The case where \(\sigma_2 \leqslant\sigma_c \leqslant\sigma_1\) follows easily, applying the "triangular inequality" \(|a+b|^4 \ll |a|^4 + |b|^4\).