# Large Deviation Estimates of Selberg’s Central Limit Theorem
and Applications

Louis-Pierre Arguin  Department of Mathematics, Baruch College, CUNY, New
York, NY Department of Mathematics, CUNY Graduate Center, New York, NY  Emma
Bailey  Department of Mathematics, CUNY Graduate Center, New York, NY

(February 20, 2022)

###### Abstract

For $V\sim\alpha\log\log T$ with $0<\alpha<2$, we prove

$\frac{1}{T}\text{meas}\\{t\in[T,2T]:\log|\zeta(1/2+{\rm
i}t)|>V\\}\ll\frac{1}{\sqrt{\log\log T}}e^{-V^{2}/\log\log T}.$

This improves prior results of Soundararajan and of Harper on the large
deviations of Selberg’s Central Limit Theorem in that range, without the use
of the Riemann hypothesis. The result implies the sharp upper bound for the
fractional moments of the Riemann zeta function proved by Heap, Radziwiłł and
Soundararajan. It also shows a new upper bound for the maximum of the zeta
function on short intervals of length $(\log T)^{\theta}$, $0<\theta<3$, that
is expected to be sharp for $\theta>0$. Finally, it yields a sharp upper bound
(to order one) for the moments on short intervals, below and above the
freezing transition. The proof is an adaptation of the recursive scheme
introduced by Bourgade, Radziwiłł and one of the authors to prove fine
asymptotics for the maximum on intervals of length $1$.

##  1 Introduction

###  1.1 Main Result

Selberg’s Central Limit Theorem [Sel46, Sel92] states that the logarithm of
the Riemann zeta function $\zeta(s)$ at a typical point on the critical line
${\rm Re}\ s=1/2$ behaves like a complex Gaussian random variable of mean $0$
and variance $\log\log T$. Specifically, if $\tau$ is uniformly distributed on
$[T,2T]$, then for the real part of the logarithm we have

$\mathbf{P}\Big{(}\log|\zeta(1/2+{\rm i}\tau)|>\sqrt{\tfrac{1}{2}\log\log
T}\cdot y\Big{)}\sim\int_{y}^{\infty}\frac{e^{-z^{2}/2}}{\sqrt{2\pi}}{\rm
d}z,\quad y\in\mathbb{R},\text{ as $T\to\infty$}.$

See [RS17] for an elegant self-contained proof of this, and [Sou21] for a
survey on the distribution of values of $L$-functions in general. In this
paper, we prove that the above Gaussian decay persists in the large deviation
regime:

######  Theorem 1.1.

Let $V\sim\alpha\log\log T$ with $0<\alpha<2$. We have for $T$ large enough

$\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|>V)\ll\frac{1}{\sqrt{\log\log
T}}\exp\left(\frac{-V^{2}}{\log\log T}\right).$

The implicit constant in the inequality can be taken uniform in $\alpha$ in
any compact subset of $(0,2)$.

Throughout the paper, the notation $\ll$ means that the left is
$\operatorname{O}$ of the right side as $T\to\infty$, and that the implicit
constant is possibly $\alpha$-dependent.

In the interval $0<V<2\log\log T$, Theorem 1.1 is an improvement of a more
general theorem of Soundararajan [Sou09], which states for this particular
range that

$\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|>V)\ll(\log T)^{{\rm
o}(1)}\cdot\exp\left(\frac{-V^{2}}{\log\log T}\right).$ (1)

Harper [Har13] also proved sharp bounds for the moments of the zeta function,
which by Markov’s inequality imply

$\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|>V)\ll\exp\left(\frac{-V^{2}}{\log\log
T}\right).$ (2)

Both results assume the Riemann hypothesis, whereas Theorem 1.1 is
unconditional. Equations (1) and (2) do hold conditionally on a wider range of
$V$, for example $V\sim k\log\log T$ for any $k>0$.

Heap, Radziwiłł and Soundararajan proved sharp upper bounds for the moments
between $0$ and $4$, cf. Corollary 1.2, which imply Equation (2)
unconditionally. For $\sqrt{\log\log T}\log\log\log T\leq V\leq 2\log\log
T-2\sqrt{\log\log T}\log\log\log T$, Heap and Soundararajan [HS20] also proved
unconditionally the asymptotic behavior

$\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|>V)=\exp\left(\frac{-V^{2}}{\log\log
T}+\operatorname{O}\left(\frac{V\log\log\log T}{\sqrt{\log\log
T}}\right)\right).$

It was conjectured by Radziwiłł [Rad11] that the Gaussian behavior actually
extends to the whole range $V\sim k\log\log T$, $k>0$, up to a multiplicative
factor

$\mathbf{P}\Big{(}\log|\zeta(1/2+{\rm i}\tau)|>V\Big{)}\sim
C_{k}\int_{V}^{\infty}\frac{e^{-y^{2}/\log\log T}}{\sqrt{\pi\log\log T}}{\rm
d}y,$

where $C_{k}$ is the conjectured leading coefficient of the $2k$-moment (cf.
[KS00]). If we write $V=\alpha\log\log T+\sigma y$ for $\sigma={\rm
o}(\sqrt{\log\log T})$, then Theorem 1.1 also gives an upper bound to order
one for a local version of Selberg’s Central Limit Theorem, as proposed in
[DBMN19]. (See Proposition 4.8 there for a more precise result for a random
model of zeta.) Finally, we also remark that for characteristic polynomials of
random unitary matrices, large deviations in the equivalent regime to Theorem
1.1 were proved in [HKO01] and precise asymptotics (including the constant)
were proved in [FMN16].

Theorem 1.1 is proved in Section 2. The method is an adaptation of a recursive
scheme introduced in [ABR20] to prove a sharp upper bound to the Fyodorov-
Hiary-Keating Conjecture, cf. Equation (7). Consider the Dirichlet polynomials

$S_{k}=\sum_{\log{2}\leq\log{p}\leq e^{k}}\frac{{\rm Re}\ p^{-{\rm
i}\tau}}{p^{1/2}},\quad k\geq 1.$ (3)

These partial sums are a good proxy for $\log|\zeta(1/2+{\rm i}\tau)|$ for $k$
close to $\log\log T$. Moreover, the moments of $S_{k}$ are very close to
Gaussian, see for example Lemma A.2 or [ABB+19, Lemma 3.4]. However, the error
for these moments is too large to handle simultaneously $k$ close to $\log\log
T$ as well as moments of order $\log\log T$.

The idea is to restrict the estimate of the probability to good events where
the partial sums (3) takes values in a narrow interval. The implementation of
this recursive scheme is much simpler here than in [ABR20], where restrictions
at every $k$ were needed. Namely, for Theorem 1.1, the partial sums only need
to be constrained on a sparse collection of $k$’s of the form

$t_{\ell}=\log\log T-\mathfrak{s}\log_{\ell+2}T,\quad\ell\geq 1,$ (4)

for some ($\alpha$-dependent) $\mathfrak{s}$, where $\log_{\ell}$ stands for
the logarithm iterated $\ell$ times. Moreover, since Theorem 1.1 only concerns
large values of $\zeta$ at a single point, no discretization is needed here
compared to [ABR20] where the authors considered the maximum of $\zeta$ over a
short $\operatorname{O}(1)$-range. This simplifies the statements and proofs
of various foundational results (cf. Lemmas 2.4, 2.6, and 2.7) regarding
second and twisted fourth moments of Dirichlet polynomials. As a corollary to
Theorem 1.1, we prove an upper bound on the maximum of $\zeta$ over a growing
window, cf. Corollary 1.3.

The restriction is on good events of the form

$\\{S_{t_{\ell}}\in[L_{\ell},U_{\ell}]\\},\quad\ell\geq 1,$

where $L_{\ell}$ is slightly below the linear interpolation $\alpha t_{\ell}$
and $U_{\ell}$ is slightly above. These barriers must be chosen carefully and
dependent on $\alpha$. Also, $U_{\ell}$ must be much higher than the upper
barrier picked in [ABR20] as the fluctuations here can be greater. It turns
out that the dominant term of the probability in Theorem 1.1 comes from the
intersection of all the good events above. On these events, the increments
$S_{t_{\ell+1}}-S_{t_{\ell}}$ are restricted to a range where large deviations
can be estimated.

Theorem 1.1 must be restricted to $\alpha<2$ since we rely on a twisted fourth
moment estimate (Lemma 2.10). More generally, large deviations in the range
$\alpha\log\log T$ are controlled by the $2\alpha$-moment of zeta. This
suggests that the method of proof should be adaptable to prove Theorem 1.1 for
any $\alpha>0$ assuming the Riemann hypothesis, where all such moments can be
sharply bounded. This would improve the bounds (1) and (2) in the full range
$\alpha\log\log T$, $\alpha>0$, conditionally. We also expect that a matching
lower bound (up to constant) can be found using the techniques of [ABR22]. In
[Rad11], it was proved that Selberg’s theorem holds up to $V$ of the order of
$(\log\log T)^{3/5-\varepsilon}$. Subsequently, Inoue [Ino19] improved the
range of $V$ up to $(\log\log T)^{2/3}$. The techniques involved in the proof
of Theorem 1.1 do not seem to be applicable to the range $V={\rm o}(\log\log
T)$. Interestingly, this leaves a gap between $V\ll(\log\log T)^{2/3}$ and
$V\sim\alpha\log\log T$ where the Gaussian decay remains open.

###  1.2 Applications

The first corollary of Theorem 1.1 is an alternative proof of a sharp upper
bound for fractional moments of the zeta function, proved unconditionally by
Heap, Radziwiłł and Soundararajan.

######  Corollary 1.2 (Theorem 1 in [HRS19]).

Let $0<\beta<4$. We have for $T$ large enough

$M_{\beta}=\frac{1}{T}\int_{T}^{2T}|\zeta(1/2+{\rm i}t)|^{\beta}{\rm
d}t\ll(\log T)^{\beta^{2}/4},$ (5)

where the implicit constant depends on $\beta$.

The proof in [HRS19] depends on twisted fourth moment estimates, as for
Theorem 1.1. Hence, it might be considered at the same conceptual level as the
proof of Corollary 1.2. Corollary 1.2 is proved in Section 3.1. Note that, via
Markov’s inequality, Equation (5) shows in particular the Gaussian decay (2)
unconditionally, for $V\sim\frac{\beta}{2}\log\log T$ and $\beta\in[0,4]$.

In short intervals, of size $(\log T)^{\theta}$ for $0\leq\theta<3$, Theorem
1.1 implies an upper bound for the maximum up to order one precision:

######  Corollary 1.3.

Let $0\leq\theta<3$ and $y>0$ such that $y={\rm o}\left(\frac{\log\log
T}{\log\log\log T}\right)$. We have

$\max_{|h|\leq(\log T)^{\theta}}|\zeta(1/2+{\rm i}t+{\rm i}h)|\leq
e^{y}\frac{(\log T)^{\sqrt{1+\theta}}}{(\log\log T)^{1/(4\sqrt{1+\theta})}},$
(6)

for all $t\in[T,2T]$ except on a set of Lebesgue measure $\ll
e^{-2\sqrt{1+\theta}y}e^{-y^{2}/\log\log T}$.

The restriction to $\theta<3$ is due to the limitations in the range of large
deviation, up to $2\log\log T$, in Theorem 1.1. The result also gives a
precise decay for the right tail of the maximum, which is exponential for
small $y$’s and Gaussian for large ones. The condition on the size of $y$ in
the statement of the corollary can be relaxed at the expense of a different
decay rate, as can be easily observed within the proof. Upper and lower bounds
for the maximum with error $(\log T)^{\varepsilon}$ were proved in [AOR19].
Corollary 1.3 proves the fine asymptotics up to order one as given in
Conjecture 1.3 of [AOR19]. The proof of Corollary 1.3 is given in Section 3.2.
It is a simple union bound after suitably discretizing the interval on $(\log
T)^{1+\theta}$ points. It is expected that the bound is sharp for $\theta>0$,
see [AAB+21] for numerical evidence of this. This is because for $\theta>0$,
the values of zeta at the $(\log T)^{1+\theta}$ points should each behave like
IID Gaussians of variance $\frac{1}{2}\log\log T$, see for example [AOR19]
111Closely related is a class of models called ‘continuous random energy
models’, cf. [BKL02, Bov06, Bov17, BH15] that exhibit similar extreme value
statistics for a suitable choice of parameters.. This is in contrast with the
case $\theta=0$. Corollary 1.3 holds for this case, but it is not sharp. It
was conjectured by Fyodorov, Hiary & Keating and Fyodorov & Keating, that the
maximum of $\log|\zeta|$ on intervals of size one should behave exactly like
the maximum of log-correlated stochastic processes [FHK12, FK14]. It was shown
in [ABR20] that

$\max_{|h|\leq 1}|\zeta(1/2+{\rm i}t+{\rm i}h)|\leq e^{y}\frac{\log
T}{(\log\log T)^{3/4}},$ (7)

for all $t\in[T,2T]$ except on a set of Lebesgue measure $\ll
ye^{-2y}e^{-y^{2}/\log\log T}$. Upper and lower bounds with error $(\log
T)^{\varepsilon}$ were proved in [Naj18, ABB+19]. A hybrid regime
interpolating between IID and log-correlated statistics was also proposed in
[ADH21]. For more on recent developments in extreme values of log-correlated
processes, see for example [BK22].

Theorem 1.1 can also be applied to improve current bounds for the moments of
$\zeta$ in short intervals.

######  Corollary 1.4.

Let $0\leq\theta<3$. For all $\beta\geq 0$, we have for $A>1$

$\int_{|h|\leq(\log T)^{\theta}}|\zeta(1/2+{\rm i}t+{\rm i}h)|^{\beta}{\rm
d}h\leq A(\log T)^{\frac{\beta^{2}}{4}+\theta},$ (8)

for all $t\in[T,2T]$ except possibly on a subset of Lebesgue measure $\ll
1/A$.

For $\beta>\beta_{c}=2\sqrt{1+\theta}$, a sharper bound holds:

$\int_{|h|\leq(\log T)^{\theta}}|\zeta(1/2+{\rm i}t+{\rm i}h)|^{\beta}{\rm
d}h\leq C_{A,\beta}\cdot(\log\log T)^{-\tfrac{\beta}{2\beta_{c}}}\cdot(\log
T)^{\tfrac{\beta_{c}}{2}\beta-1},$ (9)

for all $t\in[T,2T]$ except possibly on a subset of Lebesgue measure $\ll
1/A$, where $C_{A,\beta}$ is an explicit constant dependent on $A$ and
$\beta$.

Equation (8) was proved in [AOR19]. It follows easily by Markov’s inequality
and the bound (5). Nevertheless, we provide another proof of this using the
Lebesgue measure of high points. This is helpful in understanding the proof of
the sharper bound for the moments above $\beta_{c}$. Equation (9) is an
improvement on [AOR19], where the result was given with a $(\log
T)^{\varepsilon}$ error. Interestingly, Equation (9) is exactly the behavior
expected for the moments of $(\log T)^{1+\theta}$ IID Gaussian random
variables of variance $\frac{1}{2}\log\log T$ as computed by Bovier, Kurkova &
Löwe [BKL02, Theorem 1.6] for large $\beta$.

Equations (8) and (9) exhibit a freezing transition (also referred to as
intermittency) where the moments transition from quadratic to linear growth.
In view of this, it is natural to ask if the bound (8) at criticality
$\beta=\beta_{c}$ is sharp. At $\theta=0$, where the system seems to behave
like a log-correlated process, it can be improved as shown by Harper:

######  Theorem 1.5 (Theorem 1 and Corollary 1 in [Har19]).

We have

$\int_{|h|\leq 1}|\zeta(1/2+{\rm i}t+{\rm i}h)|^{2}{\rm d}h\leq A\frac{\log
T}{\sqrt{\log\log T}},$

for all $t\in[T,2T]$ except possibly on a subset of Lebesgue measure
$\ll\frac{(\log A)\wedge\sqrt{\log\log T}}{A}$.

The presence of the correction $1/\sqrt{\log\log T}$ is related to the
phenomenon of critical Gaussian multiplicative chaos, see [Pow18]. In Section
4, we explain how this correction appears in view of the Lebesgue measure of
high points. For $\theta>0$, where the IID heuristic prevails, such a
correction should be absent as predicted by Theorem 1.6 (i) of [BKL02]. Hence,
Equation (8) is expected to be sharp to order one at $\beta=\beta_{c}$.

Notation. Throughout the proofs, we use the probabilistic convention for
random variables and often drop the dependence on $\tau$, which will always be
taken uniform on $[T,2T]$, to lighten the notation. Most dramatically, we will
simply write

$\zeta$ for the random variable $\zeta(1/2+{\rm i}\tau)$.

Another convenient notation is

$t=\log\log T.$

It turns out that $\log\log$ is the correct scale for the primes in the
considered problems. This is because the Dirichlet sums considered, see for
example (3) and (10) below, behave like a random walk on that scale.

Acknowledgements We thank Paul Bourgade and Maksym Radziwiłł for insightful
discussions on the subject. The research of LPA was supported in part by NSF
CAREER. DMS-1653602. Part of this work was conducted whilst EB participated in
a program during the Fall 2021 semester hosted by the Mathematical Sciences
Research Institute in Berkeley, California, which was supported by the NSF
Grant No. DMS-1928930.

##  2 Proof of Theorem 1.1

The proof is an adaptation of the recursive scheme of [ABR20]. First, we
introduce some notations. Consider the partial Dirichlet sums

$S_{k}=\sum_{2\leq p\leq\exp(e^{k})}\frac{{\rm Re}\ p^{-{\rm
i}\tau}}{p^{1/2}}+\frac{{\rm Re}\ p^{-2{\rm i}\tau}}{2p},\quad k\geq 1,$ (10)

with $S_{0}=0$. (As opposed to the simpler Equation (3), we include here the
square of primes within the definition. This simplifies the application of
Lemma 2.5 below.) For $S_{k}$ to be a good approximation for $\log|\zeta|$,
the parameter $k$ must be taken close to $t$. With this in mind, $t$ is
approached in a finite number of steps by iterated logarithms as in (4):

$t_{\ell}=t-\mathfrak{s}\log_{\ell}t,\quad\ell\geq 1,$ (11)

with the convention that $t_{0}=0$. The parameter $\mathfrak{s}$ here depends
on $\alpha$. A good choice (reflecting the symmetry in $\alpha$) is

$\mathfrak{s}=\frac{2\cdot 10^{6}}{(2-\alpha)^{2}\alpha^{2}}.$ (12)

We will say more on this choice below Equation (19). The last $\ell$, denoted
by $\mathcal{L}$, is defined as the largest $\ell$ such that

$\exp(10^{6}(t-t_{\ell})^{10^{5}}e^{t_{\ell+1}})\leq\exp\Big{(}\frac{1}{100}e^{t}\Big{)}=T^{1/100}.$
(13)

Note that the left-hand side is

$\exp\Big{(}10^{6}(\mathfrak{s}\log_{\ell}t)^{10^{5}}\cdot\frac{e^{t}}{(\log_{\ell}t)^{\mathfrak{s}}}\Big{)},$

therefore the choice of $\mathfrak{s}$ ensures that such a $\mathcal{L}$
exists if $T$ is large enough. By definition, we also have
$\log_{\mathcal{L}}t=\operatorname{O}(1)$ and $\log_{\mathcal{L}}t>0$. The
corresponding complex partial sums are also needed and are denoted by

$\widetilde{S}_{k}=\sum_{2\leq p\leq\exp(e^{k})}\frac{\ p^{-{\rm
i}\tau}}{p^{1/2}}+\frac{\ p^{-2{\rm i}\tau}}{2p},\quad k\geq 1,$ (14)

and $\widetilde{S}_{0}=0$. We stress that only the values of the partial sums
at $t_{\ell}$, $1\leq\ell\leq\mathcal{L}$, are necessary.

To approximate $\exp(-S_{t_{\ell}})$, we use the mollifiers:

$\mathcal{M}_{\ell}=\sum_{\begin{subarray}{c}p|m\implies\log\log
p\in(t_{\ell-1},t_{\ell}]\\\
\Omega_{\ell}(m)\leq(t_{\ell}-t_{\ell-1})^{10^{5}}\end{subarray}}\frac{\mu(m)}{m^{\frac{1}{2}+i\tau}},$
(15)

where $\Omega_{\ell}(m)$ is the number of prime factors of $m$ in
$(\exp(e^{t_{\ell}}),\exp(e^{t_{\ell+1}})]$ with multiplicity, and $\mu(m)$ is
the Möbius function. The proof will show that product
$\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}$ is typically a good approximation
for $\exp(-S_{t_{\ell}})$.

The idea of the proof is to partition the event

$H=\\{\log|\zeta(1/2+{\rm i}\tau)|>V\\}$

into recursively defined events that greatly restrict the values of the
Dirichlet sums (10) and (14). It is expected that, if $\log|\zeta(1/2+{\rm
i}\tau)|>V$ and $V\sim\alpha t$, then the partial sum $S_{t_{\ell}}$ should be
close to $\kappa t_{\ell}$ where

$\kappa=\frac{V}{t}\sim\alpha.$ (16)

More precisely, consider for $1\leq\ell\leq\mathcal{L}$, the decreasing events

$\displaystyle A_{\ell}$
$\displaystyle=A_{\ell-1}\cap\\{|\widetilde{S}_{t_{\ell}}-\widetilde{S}_{t_{\ell-1}}|\leq\mathcal{A}(t_{\ell}-t_{\ell-1})\\}$
(17) $\displaystyle B_{\ell}$
$\displaystyle=B_{\ell-1}\cap\\{S_{t_{\ell}}\leq\kappa
t_{\ell}+\mathcal{B}\log_{\ell}t\\}$ $\displaystyle C_{\ell}$
$\displaystyle=C_{\ell-1}\cap\\{S_{t_{\ell}}\geq\kappa
t_{\ell}-\mathcal{C}\log_{\ell}t\\}$ $\displaystyle D_{\ell}$
$\displaystyle=D_{\ell-1}\cap\\{|\zeta e^{-S_{t_{\ell}}}|\leq
c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|+e^{-\mathcal{D}(t-t_{\ell-1})}\\},$

where $c_{\ell}=\prod_{j=1}^{\ell}(1+e^{-t_{j-1}})$, and
$A_{0},B_{0},C_{0},D_{0}=[T,2T]$ (the full sample space). The parameters
$\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D}$ will be chosen carefully as
discussed below. For now, we simply observe that on the good event

$G_{\ell}=A_{\ell}\cap B_{\ell}\cap C_{\ell}\cap D_{\ell},$

the partial sums are restricted in a narrow corridor between an upper and
lower barrier:

$U_{\ell}=\kappa t_{\ell}+\mathcal{B}\log_{\ell}t\qquad L_{\ell}=\kappa
t_{\ell}-\mathcal{C}\log_{\ell}t.$ (18)

The auxiliary event $D_{\ell}$ ensures that $\exp(-S_{t_{\ell}})$ is well
approximated by the mollifier, and $A_{\ell}$ is an a priori estimate needed
for the estimates involving $C_{\ell}$ and $D_{\ell}$. The probability of
$H=\\{\log|\zeta(1/2+{\rm i}\tau)|>V\\}$ can then be decomposed over the
$G_{\ell}$’s. The dominant contribution comes from $H\cap G_{\mathcal{L}}$
where the sums are restricted up to order one away from $t$. The precise
estimates are:

######  Proposition 2.1.

Let $V\sim\alpha t$ with $0<\alpha<2$. With the notation above, we have for
some $\delta>0$ (dependent on $\alpha$) and $t$ large enough

$\mathbf{P}(H\cap G_{1}^{c})\ll\frac{e^{-V^{2}/t}}{\sqrt{t}}\cdot
t^{-\delta}.$

######  Proposition 2.2.

Let $V\sim\alpha t$ with $0<\alpha<2$. With the notation above, we have for
$1\leq\ell\leq\mathcal{L}-1$, some $\delta>0$ (dependent on $\alpha$ but not
$\ell$), and $t$ large enough

$\mathbf{P}(H\cap G_{\ell}\cap
G_{\ell+1}^{c})\ll\frac{e^{-V^{2}/t}}{\sqrt{t}}\cdot(\log_{\ell}t)^{-\delta}.$

######  Proposition 2.3.

Let $V\sim\alpha t$ with $0<\alpha<2$. With the notation above, we have for
$t$ large enough

$\mathbf{P}(H\cap G_{\mathcal{L}})\ll\frac{1}{\sqrt{t}}e^{-V^{2}/t}.$

The theorem is a simple consequence of the three propositions.

###### Proof of Theorem 1.1.

It suffices to notice that

$\mathbf{P}(H)=\mathbf{P}(H\cap
G_{1}^{c})+\sum_{\ell=1}^{\mathcal{L}-1}\mathbf{P}(H\cap G_{\ell}\setminus
G_{\ell+1})+\mathbf{P}(H\cap G_{\mathcal{L}}).$

The result follows by applying Propositions 2.1, 2.1, 2.3. ∎

As mentioned above, the parameters in (17) need to be chosen in a delicate
manner. As we shall see from the proof (cf. Equations (29) and (35)), the
choice of $\mathcal{B}$ must satisfy the following restrictions.

$\displaystyle 1+\alpha^{2}\mathfrak{s}-2\alpha\mathcal{B}$ $\displaystyle<0$
(19) $\displaystyle\mathcal{B}-\alpha\mathfrak{s}$ $\displaystyle<0.$

The first equation forces $\mathcal{B}$ to be proportional to $1/\alpha$ to
handle small $\alpha$’s. In turn, the second equation leads to
$\mathfrak{s}>1/\alpha^{2}$, motivating in part the choice of $\mathfrak{s}$
in (12). With this choice, the defining inequalities for $\mathcal{B}$ becomes

$\displaystyle\frac{1}{2\alpha}+\frac{10^{6}}{\alpha(2-\alpha)^{2}}<$
$\displaystyle\mathcal{B}<\frac{10^{6}}{\alpha(2-\alpha)^{2}}+\frac{10^{6}}{\alpha(2-\alpha)^{2}}.$

This is a non-empty interval since $\alpha>0$. Therefore, a valid choice is

$\mathcal{B}=\frac{3\cdot 10^{6}}{2\alpha(2-\alpha)^{2}}+\frac{1}{4\alpha}.$
(20)

The restrictions on $\mathcal{C}$ (cf. Equations (32) and (38)) will be

$\mathcal{C}>\frac{1}{2(2-\alpha)}\Big{\\{}1+(2-\alpha)^{2}\mathfrak{s}\Big{\\}}.$
(21)

(We note in passing that this is the first constraint for $\mathcal{B}$ in
(19), after the transformation $\alpha\mapsto 2-\alpha$.) Therefore, a valid
choice for $\mathcal{C}$ is

$\mathcal{C}=\frac{3\cdot
10^{6}}{2\alpha^{2}(2-\alpha)}+\frac{1}{4(2-\alpha)}.$ (22)

This choice implies the upper bound $\mathcal{C}<(2-\alpha)\mathfrak{s}$.

The parameter $\mathcal{A}$ will need to satisfy (cf. Equations (34) and
(42)):

$\mathcal{A}>\frac{\alpha^{2}}{4}+\frac{\alpha\mathcal{C}}{2\mathfrak{s}}+2.$
(23)

This choice implies in particular

$\mathcal{A}^{2}>\alpha^{2}+\frac{2\alpha\mathcal{C}}{\mathfrak{s}}+4.$ (24)

For example, one can take

$\mathcal{A}=10^{3},$ (25)

since, with the choices of $\mathcal{C}$ and $\mathfrak{s}$ above, and for
$0<\alpha<2$, we have

$\displaystyle\mathcal{A}=10^{3}$
$\displaystyle>4+\frac{\alpha(2-\alpha)}{8}(\alpha^{2}+3)$ (26)
$\displaystyle>\frac{\alpha^{2}}{4}+\frac{\alpha\mathcal{C}}{2\mathfrak{s}}+2.$

Finally, the conditions on $\mathcal{D}$ will be as in [ABR20]

$\mathcal{D}=10^{4}.$ (27)

###  2.1 Proof of Proposition 2.1

First, notice that

$H\cap G_{1}^{c}\subset\ \ A_{1}^{c}\ \cup\ B_{1}^{c}\ \cup\ (H\cap
C_{1}^{c}\cap A_{1}\cap D_{1})\ \cup\ (D_{1}^{c}\cap A_{1}).$

We estimate the probability of the four events in the union on the right
individually.

We first evaluate $A_{1}^{c}$:

$\mathbf{P}(A_{1}^{c})=\mathbf{P}(|\widetilde{S}_{t_{1}}|>\mathcal{A}t_{1}).$

Equation (77) of the appendix is applicable with the choice $q=\lceil
2\mathcal{A}^{2}t_{1}\rceil$, and implies that this is

$\ll\sqrt{t_{1}}\cdot\exp(-\mathcal{A}^{2}t_{1}).$

Since $\mathcal{A}=10^{3}$, for some $\delta>0$ this is clearly

$\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}t^{-\delta}.$ (28)

Turning to $B_{1}^{c}$, and applying Markov’s inequality for some $q>1$ yields

$\mathbf{P}(B_{1}^{c})\leq\mathbf{P}(S_{t_{1}}>U_{1})\leq
U_{1}^{-2q}\mathbf{E}[|S_{t_{1}}|^{2q}].$

Equation (78) then applies with $q=\lceil U_{1}^{2}/t_{1}\rceil$, giving

$\mathbf{P}(B_{1}^{c})\ll\sqrt{t_{1}}e^{-U_{1}^{2}/t_{1}}\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot
t^{1+\kappa^{2}\mathfrak{s}-2\kappa\mathcal{B}}.$ (29)

By the choice of $\mathcal{B}$ in Equation (19), one has
$1+\alpha^{2}\mathfrak{s}-2\alpha\mathcal{B}<0$. Since $\kappa=\alpha+{\rm
o}(1)$ by Equation (16), the above is

$\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot t^{-\delta}$ (30)

for some $\delta>0$, depending on $\alpha$ and different from (28).

To evaluate $H\cap C_{1}^{c}\cap A_{1}\cap D_{1}$, we require the following
lemma, proved in Section 2.5.

######  Lemma 2.4.

For $w$ with $|w|\leq 4t_{1}$, we have

$\mathbf{E}\left[|\zeta\mathcal{M}_{1}|^{4}\
\mathbf{1}\left(S_{t_{1}}\in(w,w+1]\right)\right]\ll
e^{4(t-t_{1})}\cdot\frac{e^{-w^{2}/t_{1}}}{\sqrt{t}}.$ (31)

Let us explain the intuition behind the result. One should think of
$\zeta\mathcal{M}_{1}$ as a random Euler product involving primes larger than
$\exp(e^{t_{1}})$. Furthermore, Selberg’s result suggests its logarithm should
be distributed like a Gaussian random variable of variance $t-t_{1}$. This
explains the first factor $e^{4(t-t_{1})}$ as the contribution from the moment
generating function of such a variable. As explained in Section 2.4, the
indicator function can be approximated by a suitable Dirichlet polynomial
involving primes less than $S_{t_{1}}$. Since primes should behave
independently, it is not surprising to see the decoupling between the factors.
Most importantly, we obtain a Gaussian behavior for the variable $S_{t_{1}}$
in a large deviation regime.

The estimate $\mathbf{P}(H\cap C_{1}^{c}\cap A_{1}\cap D_{1})$ is done by
first partitioning on the value of $S_{t_{1}}$ using the restrictions given by
$A_{1}$ and $C_{1}^{c}$:

$\displaystyle\mathbf{P}(H\cap C_{1}^{c}\cap A_{1}\cap D_{1})$
$\displaystyle\leq\sum_{-\mathcal{A}(t_{1}-t_{0})<u<L_{1}}\mathbf{P}(\\{S_{t_{1}}\in(u,u+1],|\zeta|>e^{V}\\}\cap
D_{1})$
$\displaystyle\leq\sum_{-\mathcal{A}(t_{1}-t_{0})<u<L_{1}}\mathbf{P}(\\{S_{t_{1}}\in(u,u+1],|\zeta
e^{-S_{t_{1}}}|>e^{V-u-1}\\}\cap D_{1}),$

where we recall that $V=\kappa t$. The event $D_{1}$ implies that

$|\zeta e^{-S_{t_{1}}}|\leq
2|\zeta\mathcal{M}_{1}|+e^{-\mathcal{D}(t-t_{0})}.$

Therefore if $|\zeta e^{-S_{t_{1}}}|>e^{V-u-1}$, then it must be that either

$2|\zeta\mathcal{M}_{1}|>\frac{1}{2}e^{V-u-1}$

or

$e^{-\mathcal{D}(t-t_{0})}>\frac{1}{2}e^{V-u-1}.$

The latter case is impossible, since it the exponent on the left side is
negative, whereas on the right side we have on the range of $u$

$V-u-1>\kappa t-\kappa t_{1}+\mathcal{C}\log t-1>0.$

This implies that

$\displaystyle\mathbf{P}(A_{1}\cap D_{1}\cap H\cap C_{1}^{c})$
$\displaystyle\leq\sum_{-\mathcal{A}(t_{1}-t_{0})<u<L_{1}}\mathbf{P}(|\zeta\mathcal{M}_{1}|>\frac{1}{100}e^{V-u}\cap\\{S_{t_{1}}\in(u,u+1]\\})$
$\displaystyle\ll\sum_{u<L_{1}}e^{-4(V-u)}\cdot\mathbf{E}[|\zeta\mathcal{M}_{1}|^{4}{\mathbf{1}}(S_{t_{1}}\in(u,u+1])].$

The sum over $u<0$ is $\ll e^{-4V}\cdot e^{4(t-t_{1})}$ which is much smaller
than $\frac{e^{-\kappa^{2}t}}{\sqrt{t}}t^{-\delta}$ for the range of $V$
considered. Lemma 2.4 can be applied on the range $0\leq u<L_{1}$ This gives

$\ll
e^{4(t-t_{1})}\sum_{u<L_{1}}e^{-4(V-u)}\frac{e^{-u^{2}/t_{1}}}{\sqrt{t_{1}}}.$

After the change of variable $w=\kappa t_{1}-u$, this becomes

$\displaystyle e^{(4-4\kappa)(t-t_{1})}\sum_{u<L_{1}}e^{-4(\kappa
t_{1}-u)}\frac{e^{-u^{2}/t_{1}}}{\sqrt{t_{1}}}$
$\displaystyle\ll\frac{e^{-\kappa^{2}t_{1}}}{\sqrt{t_{1}}}e^{(4-4\kappa)(t-t_{1})}\sum_{w>\mathcal{C}\log
t}e^{-(4-2\kappa)w}$ (32)
$\displaystyle\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot
t^{\mathfrak{s}\kappa^{2}+\mathfrak{s}(4-4\kappa)-2(2-\kappa)\mathcal{C}}$
$\displaystyle\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}t^{-\delta},$

for some $\delta>0$, by the choice of $\mathcal{C}$ in (21).

Finally we estimate $D_{1}^{c}$. In order to proceed, we need the following
lemma. The proof follows by expressing $e^{-(S_{t_{\ell+1}}-S_{t_{\ell}})}$ in
terms of an Euler product, and by bounding the contribution of integers $m$
with $\Omega_{\ell}(m)>(t_{\ell}-t_{\ell-1})^{10^{5}}$ using Rankin’s trick.

######  Lemma 2.5 (Lemma 23 in [ABR20]).

Suppose $\ell\geq 0$ and that
$|\widetilde{S}_{t_{\ell+1}}-\widetilde{S}_{t_{\ell}}|\leq
10^{3}(t_{\ell+1}-t_{\ell})$. Then we have

$e^{-(S_{t_{\ell+1}}-S_{t_{\ell}})}\leq(1+e^{-t_{\ell}})|\mathcal{M}_{\ell+1}|+e^{-10^{5}(t_{\ell+1}-t_{\ell})}.$

Now, observe that the event $A_{1}\cap\\{|\zeta|\leq e^{2t}\\}$ is contained
in $A_{1}\cap D_{1}$. Indeed, since
$|\widetilde{S}_{t_{1}}-\widetilde{S}_{t_{0}}|\leq 10^{3}(t_{1}-t_{0})$ on
$A_{1}$, Lemma 2.5 implies

$|\zeta e^{-(S_{t_{1}}-S_{t_{0}})}|\leq
2|\zeta\mathcal{M}_{1}|+|\zeta|e^{-10^{5}(t_{1}-t_{0})}\leq
2|\zeta\mathcal{M}_{1}|+e^{2t-10^{5}(t_{1}-t_{0})},$

which implies $D_{1}$ since $\mathcal{D}=10^{4}$. Hence, to estimate
$\mathbf{P}(D_{1}^{c}\cap A_{1})$, it suffices to estimate
$\mathbf{P}(|\zeta|>e^{2t})$:

$\mathbf{P}(D_{1}^{c}\cap A_{1})\leq\mathbf{P}(|\zeta|>e^{2t})\leq
e^{-4t}\mathbf{E}[|\zeta|^{2}]\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}e^{-100t},$
(33)

since $\mathbf{E}[|\zeta|^{2}]\ll e^{t}$ [Theorem 2.41 in [HL18]].

Summarising, we have by a union bound and successively applying Equations
(28), (30), (32), and (33),

$\displaystyle\mathbf{P}(H\cap G_{1}^{c})$
$\displaystyle\leq\mathbf{P}(A_{1}^{c})+\mathbf{P}(B_{1}^{c})+\mathbf{P}(A_{1}\cap
D_{1}\cap H\cap C_{1}^{c})+\mathbf{P}(A_{1}\cap D_{1}^{c})$
$\displaystyle\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}t^{-\delta},$

for some $\delta>0$ dependent on $\alpha$.

###  2.2 Proof of Proposition 2.2

Notice that

$H\cap G_{\ell}\cap G_{\ell+1}^{c}\subset\ (A_{\ell+1}^{c}\cap G_{\ell})\
\cup\ (B_{\ell+1}^{c}\cap G_{\ell})\ \cup\ (H\cap C_{\ell+1}^{c}\cap
A_{\ell+1}\cap D_{\ell+1}\cap G_{\ell})\ \cup\ (D_{\ell+1}^{c}\cap
A_{\ell+1}\cap G_{\ell}).$

The probability of each event in the union on the right side are now
evaluated. In order to handle the event involving $A_{\ell+1}^{c}$ we will
need the following lemma, proved in Section 2.4.

######  Lemma 2.6.

Let $\ell\geq 1$ be such that
$10^{6}(t-t_{\ell})^{10^{5}}e^{t_{\ell+1}}\leq\frac{1}{100}e^{t}$. Let
$\mathcal{Q}$ be a Dirichlet polynomial of length
$N\leq\exp(\frac{1}{100}e^{t})$, supported on integers all of whose prime
factors are greater than $\exp(e^{t_{\ell}})$. Then for
$w\in[L_{\ell},U_{\ell}]$, we have

$\mathbb{E}\left[|\mathcal{Q}(\tfrac{1}{2}+{\rm
i}\tau)|^{2}\mathbf{1}\left(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\}\right)\right]\ll\mathbb{E}\left[|\mathcal{Q}(\tfrac{1}{2}+{\rm
i}\tau)|^{2}\right]\cdot\frac{e^{-w^{2}/t_{\ell}}}{\sqrt{t_{\ell}}}.$

As in Lemma 2.7, the decoupling is due to the fact that the Dirichlet
polynomials involve primes in different intervals. Though the events
$B_{\ell}\cap C_{\ell}$ do not appear explicitly in the result, their presence
here is crucial to obtain the Gaussian behavior of $S_{t_{\ell}}$ in a large
deviation regime.

We first show that $\ell\geq 1$

$\mathbf{P}(A_{\ell+1}^{c}\cap
G_{\ell})\ll\frac{e^{-V^{2}/t}}{\sqrt{t}}\cdot(\log_{\ell}t)^{-\delta}.$

For any $q>1$, the probability $\mathbf{P}(A_{\ell+1}^{c}\cap G_{\ell})$ is
smaller than

$\displaystyle\sum_{u\in[L_{\ell},U_{\ell}]}\mathbf{E}\left[\frac{|\widetilde{S}_{t_{\ell+1}}-\widetilde{S}_{t_{\ell}}|^{2q}}{(\mathcal{A}(t_{\ell+1}-t_{\ell}))^{2q}}\mathbf{1}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(u,u+1]\\})\right].$

With the choice $q=\lceil 2\mathcal{A}^{2}(t_{\ell+1}-t_{\ell})\rceil$, the
polynomial
$\mathcal{Q}=|\widetilde{S}_{t_{\ell+1}}-\widetilde{S}_{t_{\ell}}|^{2q}$ both
satisfies the assumptions of Lemma 2.6 and Lemma A.1. Therefore, the above is

$\displaystyle\ll\sum_{u\in[L_{\ell},U_{\ell}]}(t_{\ell+1}-t_{\ell})^{1/2}e^{-{\mathcal{A}}^{2}(t_{\ell+1}-t_{\ell})}\frac{e^{-u^{2}/t_{\ell}}}{\sqrt{t_{\ell}}}$
$\displaystyle\ll(t_{\ell+1}-t_{\ell})^{1/2}\cdot
e^{-{\mathcal{A}}^{2}(t_{\ell+1}-t_{\ell})}\cdot\frac{e^{-L_{\ell}^{2}/t_{\ell}}}{\sqrt{t_{\ell}}},$

where the last inequality is by estimating the sum over $u$ trivially. Since
$L_{\ell}=\kappa t_{\ell}-\mathcal{C}\log_{\ell}t$, this is

$\displaystyle\ll\frac{e^{-\kappa^{2}t_{\ell}}}{\sqrt{t}}\cdot(\log_{\ell-1}t)^{-{\mathcal{A}}^{2}\mathfrak{s}+2\kappa\mathcal{C}}\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot(\log_{\ell-1}t)^{\kappa^{2}\mathfrak{s}-{\mathcal{A}}^{2}\mathfrak{s}+2\kappa\mathcal{C}}.$
(34)

The choice of parameters in Equation (24) guarantees that the exponent is
negative.

Now we show that for $\ell\geq 1$,

$\mathbf{P}(B_{\ell+1}^{c}\cap
G_{\ell})\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}\cdot(\log_{\ell}t)^{-\delta}.$

By partitioning on the position of $S_{t_{\ell}}$, we have

$\displaystyle\mathbf{P}(B_{\ell+1}^{c}\cap G_{\ell})$
$\displaystyle\ll\mathbf{P}(B_{\ell+1}^{c}\cap B_{\ell}\cap C_{\ell})$
$\displaystyle\ll\sum_{u\in[L_{\ell},U_{\ell}]}\mathbf{P}(\\{S_{t_{\ell+1}}-S_{t_{\ell}}>U_{\ell+1}-u\\}\cap\\{S_{t_{\ell}}\in(u,u+1]\\}\cap
B_{\ell-1}\cap C_{\ell-1})$
$\displaystyle\ll\sum_{u\in[L_{\ell},U_{\ell}]}\mathbf{E}\left[\frac{(S_{t_{\ell+1}}-S_{t_{\ell}})^{2q}}{(U_{\ell+1}-u)^{2q}}\mathbf{1}(S_{t_{\ell}}\in(u,u+1],S_{t_{k}}\in[L_{k},U_{k}]\
\forall k<\ell)\right],$

where the final line holds for any $q>1$ by an application of Markov’s
inequality, provided that $U_{\ell+1}-U_{\ell}>0$. This holds by the choice of
$\mathcal{B}$ and $\mathfrak{s}$ in Equations (12) and (19).

Choosing $q=\lceil(U_{\ell}-u)^{2}/(t_{\ell+1}-t_{\ell})\rceil$, then the
Dirichlet polynomial $(S_{t_{\ell+1}}-S_{t_{\ell}})^{q}$ has length at most
$\exp(2qe^{t_{\ell+1}})$ so the the conditions of Lemma 2.6 and Lemma A.2 are
satisfied. An application of Equation (79) then yields

$\displaystyle\mathbf{P}(B_{\ell+1}^{c}\cap G_{\ell})$
$\displaystyle\ll\sum_{u\in[L_{\ell},U_{\ell}]}e^{-\frac{(U_{\ell+1}-u)^{2}}{t_{\ell+1}-t_{\ell}}}\cdot\frac{e^{-u^{2}/t_{\ell}}}{\sqrt{t_{\ell}}}\ll\sqrt{t_{\ell+1}-t_{\ell}}\cdot\frac{e^{-U_{\ell+1}^{2}/t_{\ell}}}{\sqrt{t}}.$

The last bound follows by bounding the sum over $u$ by the Gaussian integral.
Since $U_{\ell+1}=\kappa t_{\ell+1}+\mathcal{B}\log_{\ell+1}t$, this is
bounded by

$\ll\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot(\log_{\ell}t)^{1/2+\mathfrak{s}\kappa^{2}-2\kappa\mathcal{B}}.$
(35)

The choice of $\mathcal{B}$ in Equation (19) ensures that
$1/2+\mathfrak{s}\kappa^{2}-2\kappa\mathcal{B}<0$.

The next estimate is

$\mathbf{P}(H\cap C_{\ell+1}^{c}\cap A_{\ell+1}\cap D_{\ell+1}\cap G_{\ell}).$

For this, we need a more detailed version of Lemma 2.4.

######  Lemma 2.7.

Let $\ell\geq 1$ such that
$10^{6}(t-t_{\ell})^{10^{5}}e^{t_{\ell+1}}\leq\frac{1}{100}e^{t}$. For
$w\in[L_{\ell},U_{\ell}]$, we have

$\mathbf{E}\left[|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|^{4}\
\mathbf{1}\left(B_{\ell}\cap C_{\ell},S_{t_{\ell}}\in[w,w+1]\right)\right]\ll
e^{4(t-t_{\ell})}\cdot\frac{e^{-w^{2}/t_{\ell}}}{\sqrt{t}}.$ (36)

Moreover, let $\gamma(m)$ be a sequence of complex coefficients with
$|\gamma(m)|\leq\exp(\frac{1}{1000}e^{t})$ for all $m\geq 1$. Set

$\mathcal{Q}_{\ell}=\sum_{\begin{subarray}{c}p|m\implies\log\log
p\in(t_{\ell},t_{\ell+1}]\\\
\Omega_{\ell+1}(m)\leq(t_{\ell+1}-t_{\ell})^{10^{4}}\end{subarray}}\frac{\gamma(m)}{m^{\frac{1}{2}+{\rm
i}\tau}}.$

We have

$\mathbf{E}\left[|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|^{4}\right.\left.|\mathcal{Q}_{\ell}|^{2}\mathbf{1}\left(B_{\ell}\cap
C_{\ell},S_{t_{\ell}}\in[w,w+1]\right)\right]\ll
e^{4(t-t_{\ell+1})}\cdot\mathbb{E}\left[|\mathcal{Q}_{\ell}|^{2}\right]\cdot\frac{e^{-w^{2}/t_{\ell}}}{\sqrt{t}}.$
(37)

We now partition the values of $S_{t_{\ell}}=u$ for $L_{\ell}\leq u\leq
U_{\ell}$ (on the event $G_{\ell})$ as well as the values of the increments
$S_{t_{\ell+1}}-S_{t_{\ell}}=v$ with the restrictions $u+v<L_{\ell+1}$ (on the
event $C_{\ell+1}^{c}$) and $|v|\leq\mathcal{A}(t_{\ell+1}-t_{\ell})$ (on the
event $A_{\ell+1}$). The above is then smaller than

$\sum_{\begin{subarray}{c}u\in[L_{\ell},U_{\ell}]\\\ u+v\leq L_{\ell+1}\\\
|v|\leq\mathcal{A}(t_{\ell+1}-t_{\ell})\end{subarray}}\mathbf{P}(\\{S_{t_{\ell}}\in(u,u+1],S_{t_{\ell+1}}-S_{t_{\ell}}\in(v,v+1],|\zeta
e^{-S_{t_{\ell+1}}}|>e^{V-(u+v)}\\}\cap B_{\ell}\cap C_{\ell}\cap
D_{\ell+1}).$

The definition of the event $D_{\ell+1}$ and the fact that $|\zeta
e^{-S_{t_{\ell+1}}}|>e^{V-(u+v+1)}$ imply that either

$c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|>\frac{1}{2}e^{V-(u+v+1)},$

or

$e^{-\mathcal{D}(t-t_{\ell})}>\frac{1}{2}e^{V-(u+v+1)}.$

Again, the last case cannot occur, since the exponent on the left side is
negative whereas the one on the right is

$V-(u+v)-1>V-L_{\ell+1}-1=(\mathfrak{s}\kappa+\mathcal{C})\log_{\ell+1}t-1>0,$

for $1\leq\ell\leq\mathcal{L}-1$ since $\log_{\mathcal{L}}t>0$ by
construction. This reduces this estimate to

$\displaystyle\sum_{\begin{subarray}{c}u\in[L_{\ell},U_{\ell}]\\\ u+v\leq
L_{\ell+1}\\\
|v|\leq\mathcal{A}(t_{\ell+1}-t_{\ell})\end{subarray}}\mathbf{P}(\\{S_{t_{\ell}}\in(u,u+1],S_{t_{\ell+1}}-S_{t_{\ell}}\in(v,v+1],|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|>\frac{1}{100}e^{V-(u+v)}\\}\cap
B_{\ell}\cap C_{\ell})$
$\displaystyle\ll\sum_{\begin{subarray}{c}u\in[L_{\ell},U_{\ell}]\\\ u+v\leq
L_{\ell+1}\\\
|v|\leq\mathcal{A}(t_{\ell+1}-t_{\ell})\end{subarray}}e^{-4V+4(u+v)}\mathbf{E}\Big{[}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|^{4}\cdot\frac{|S_{t_{\ell+1}}-S_{t_{\ell}}|^{2q}}{|v|^{2q}}\mathbf{1}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(u,u+1]\\})\Big{]},$

by Markov’s inequality with
$q=\lceil|v|^{2}/(t_{\ell+1}-t_{\ell})\rceil\leq\mathcal{A}^{2}(t_{\ell+1}-t_{\ell})$.
Applications of Lemma 2.7, Equation 37 with
$\mathcal{Q}_{\ell}=(S_{t_{\ell+1}}-S_{t_{\ell}})^{q}$ and Equation (79) then
implies that this is

$\ll\sum_{\begin{subarray}{c}u\in[L_{\ell},U_{\ell}]\\\ u+v\leq L_{\ell+1}\\\
|v|\leq\mathcal{A}(t_{\ell+1}-t_{\ell})\end{subarray}}e^{-4V+4(u+v)}\cdot
e^{4(t-t_{\ell+1})}\cdot
e^{-v^{2}/(t_{\ell+1}-t_{\ell})}\frac{e^{-u^{2}/t_{\ell}}}{\sqrt{t_{\ell}}}.$

The change of variables $\bar{u}=u-\kappa t_{\ell}$ and
$\bar{v}=v-\kappa(t_{\ell+1}-t_{\ell})$ and dropping some conditions on the
sum gives

$\ll\frac{e^{-\kappa^{2}t_{\ell+1}}}{\sqrt{t}}e^{(4-4\kappa)(t-t_{\ell+1})}\cdot\sum_{\begin{subarray}{c}\bar{v}\in\mathbb{Z}\\\
\bar{u}+\bar{v}\leq-\mathcal{C}\log_{\ell+1}t\end{subarray}}e^{(4-2\kappa)(\bar{u}+\bar{v})}e^{-\bar{v}^{2}/(t_{\ell+1}-t_{\ell})},$

where we dropped the term $e^{-\bar{u}^{2}/t_{\ell}}$ since it is of order one
by the restriction on $\bar{u}$. It remains to sum over $\bar{u}+\bar{v}$
first, then do the Gaussian sum on $\bar{v}$ to get

$\frac{e^{-\kappa^{2}t}}{\sqrt{t}}\cdot(\log_{\ell}t)^{\mathfrak{s}(2-\kappa)^{2}-2(2-\kappa)\mathcal{C}+1/2}.$
(38)

Again, the last term is $(\log_{\ell}t)^{-\delta}$ by the choice of parameters
in (21).

Finally we consider $D_{\ell+1}^{c}\cap A_{\ell+1}\cap G_{\ell}$. We claim
that it is enough to evaluate

$\mathbf{P}(\\{|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|>e^{\mathcal{\mathcal{A}}(t-t_{\ell})}\\}\cap
G_{\ell}).$

To see this, it suffices to notice that
$A_{\ell+1}\cap\\{|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|\leq
e^{\mathcal{A}(t-t_{\ell})}\\}\cap D_{\ell}$ is in $A_{\ell+1}\cap
D_{\ell+1}$. Indeed, on the event
$A_{\ell+1}\cap\\{|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|\leq
e^{\mathcal{A}(t-t_{\ell})}\\}\cap D_{\ell}$, we have

$\displaystyle|S_{t_{\ell+1}}-S_{t_{\ell}}|$
$\displaystyle\leq\mathcal{A}(t_{\ell+1}-t_{\ell})$ (39) $\displaystyle|\zeta
e^{-S_{t_{\ell}}}|$ $\displaystyle\leq
c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|+e^{-\mathcal{D}(t-t_{\ell-1})}$
(40) $\displaystyle|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|$
$\displaystyle\leq e^{\mathcal{A}(t-t_{\ell})}.$ (41)

Equations (39) and (40) imply that

$\displaystyle|\zeta e^{-S_{t_{\ell+1}}}|$
$\displaystyle\leq\left(c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|+e^{-10^{4}(t-t_{\ell-1})}\right)e^{-(S_{t_{\ell+1}}-S_{t_{\ell}})}$
$\displaystyle\leq
c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell}|e^{-(S_{t_{\ell+1}}-S_{t_{\ell}})}+e^{-10^{3}(t-t_{\ell-1})},$

for $t$ large enough. Then, Lemma 2.5 gives

$|\zeta e^{-S_{t_{\ell+1}}}|\leq
c_{\ell}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|+c_{\ell}e^{10^{3}(t-t_{\ell})-10^{5}(t_{\ell+1}-t_{\ell})}+e^{-10^{3}(t-t_{\ell-1})}.$

We conclude that $D_{\ell+1}$ holds.

It remains to estimate
$\mathbf{P}(\\{|\zeta\mathcal{M}_{\ell}|>e^{\mathcal{A}(t-t_{\ell})}\\}\cap
G_{\ell})$. We have by subsequent applications of Markov’s inequality and
Lemma 2.7, Equation (36),

$\displaystyle\mathbf{P}(\\{|\zeta\mathcal{M}_{\ell}|>e^{\mathcal{A}(t-t_{\ell})}\\}\cap
G_{\ell})$ $\displaystyle\ll
e^{-4\mathcal{A}(t-t_{\ell})}\mathbf{E}[|\zeta\mathcal{M}_{\ell}|^{4}\mathbf{1}(G_{\ell})]$
$\displaystyle\ll
e^{-4(\mathcal{A}-1)(t-t_{\ell})}\frac{e^{-L_{\ell}^{2}/t_{\ell}}}{\sqrt{t_{\ell}}}$
$\displaystyle\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}e^{-4\mathfrak{s}(\mathcal{A}-1)\log_{\ell}t}e^{\mathfrak{s}\kappa^{2}\log_{\ell}t+2\kappa\mathcal{C}\log_{\ell}t}.$

We conclude that

$\mathbf{P}(D_{\ell+1}^{c}\cap A_{\ell+1}\cap
G_{\ell})\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}\cdot(\log_{\ell-1}t)^{\mathfrak{s}\kappa^{2}+2\kappa\mathcal{C}-4\mathfrak{s}(\mathcal{A}-1)}.$
(42)

The exponent is negative by Equation (23).

###  2.3 Proof of Proposition 2.3

Finally we establish that

$\mathbf{P}(H\cap G_{\mathcal{L}})\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}.$

After partitioning on the value of $S_{t_{\mathcal{L}}}$, applying Markov’s
inequality, and subsequently Lemma 2.7 we have

$\mathbf{P}(H\cap
G_{\mathcal{L}})\ll\sum_{v\in[L_{\mathcal{L}},U_{\mathcal{L}}]}e^{4(t-t_{\mathcal{L}})}e^{4v}\frac{e^{-v^{2}/t_{\mathcal{L}}}}{\sqrt{t_{\mathcal{L}}}}.$

Applying the transformation $w=v-\kappa t_{\mathcal{L}}$, the probability is
bounded by

$\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}e^{(2-\kappa)^{2}\log_{\mathcal{L}}t}\sum_{-\mathcal{C}\log_{\mathcal{L}}t<w<\mathcal{B}\log_{\mathcal{L}}t}e^{2(2-\kappa)w}.$

Since $\alpha<2$, the sum is bounded by
$\exp(2(2-\kappa)\mathcal{B}\log_{\mathcal{L}}t)$, so after grouping we find

$\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}\cdot
e^{(2-\kappa+2\mathcal{B})(2-\kappa)\log_{\mathcal{L}}t}.$

By the choice of $\mathcal{L}$, this is
$\ll\frac{1}{\sqrt{t}}e^{-\kappa^{2}t}$.

###  2.4 Proof of Lemma 2.6

We express the event $B_{\ell}\cap C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\}$
in terms of the increments

$Y_{j}=S_{t_{j}}-S_{t_{j-1}},\quad 1\leq j\leq\ell.$ (43)

The event implies that $S_{t_{j}}\in[L_{j},U_{j}]$ for all $j$. We partition
these intervals into subintervals of width $\Delta_{j}^{-1}$ where

$\Delta_{j}=(t_{j}-t_{j-1}),$

so $\Delta_{j}\leq\mathfrak{s}\log_{j-1}t$ for $j>1$, and $\Delta_{1}$ is
effectively $t_{1}=t-\mathfrak{s}\log t$. Note that $\Delta_{j}$ is of the
same order as the variance of $Y_{j}$. Moreover, we have

$\sum_{j\geq 1}\Delta_{j}^{-1}\leq 1.$

Consider the set $\mathcal{I}$ of $\ell$-tuples
$\mathbf{u}=(u_{1},\dots,u_{\ell})$ such that

$\displaystyle\sum_{i=1}^{j}u_{i}\in[L_{j}-1,U_{j}+1],\quad
j\leq\ell,\qquad\sum_{i=1}^{\ell}u_{i}\in[w-1,w+1].$ (44)

As a consequence of the definition, we have for all $j>1$

$\displaystyle L_{j}-1-(U_{j-1}+1)$ $\displaystyle\leq u_{j}\leq
U_{j}+1-(L_{j-1}-1)$

which implies
$|u_{j}|\leq(\kappa\mathfrak{s}+\mathcal{B}+\mathcal{C})\log_{j-1}t+2$. We
will also shortly require the following estimate. Since
$\mathcal{B}<\alpha\mathfrak{s}$ and $\mathcal{C}<(2-\alpha)\mathfrak{s}$ (by
(20) and (22)), we conclude from $\alpha<2$ that

$|u_{j}|<4\Delta_{j}+2.$ (45)

With these definitions, it is straightforward to check that we have the
following inclusion of events

$B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\}\subset\bigcup_{\mathbf{u}\in\mathcal{I}}\\{Y_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}],1\leq
j\leq\ell\\}.$ (46)

In particular, this implies

$\mathbf{1}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\})\leq\sum_{\mathbf{u}\in\mathcal{I}}\prod_{j}\mathbf{1}(Y_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}]).$
(47)

We first prove:

######  Lemma 2.8.

In the above notation, we have for $A\geq 10$ and $j\leq\ell$,

$\mathbf{1}(Y_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}])\leq|\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}(1+ce^{-\Delta_{j}^{A-1}}),$
(48)

where $c$ is an absolute constant and
$\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})$ is a Dirichlet polynomial on
integers $n$ whose prime factors are in $(\exp(e^{t_{j-1}}),\exp(e^{t_{j}})]$
with $\Omega(n)\leq\Delta_{j}^{10A}$. In particular, its length is less than
than $\exp(2e^{t_{j}}\Delta_{j}^{10A})$.

###### Proof.

Lemma 6 in [ABR20] states that for any $\Delta,A\geq 3$, there exists an
entire function $G_{\Delta,A}(x)\in L^{2}(\mathbb{R})$ such that for some
absolute constant $c>0$:

  1. 1.

the Fourier transform $\widehat{G}_{\Delta,A}(x)$ is supported on
$[-\Delta^{2A},\Delta^{2A}]$;

  2. 2.

$0\leq G_{\Delta,A}(x)\leq 1$ for all $x\in\mathbb{R}$;

  3. 3.

$\mathbf{1}(x\in[0,\Delta^{-1}])\leq
G_{\Delta,A}(x)\cdot(1+ce^{-\Delta^{A-1}});$

  4. 4.

$G_{\Delta,A}(x)\leq\mathbf{1}(x\in[-\Delta^{-A/2},\Delta^{-1}+\Delta^{-A/2}])+ce^{-\Delta^{A-1}};$

  5. 5.

$\int_{\mathbb{R}}|\widehat{G}_{\Delta,A}(x)|{\rm d}x\leq 2\Delta^{2A}.$

From the property (3), we get

$\mathbf{1}(Y_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}])\leq|G_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}(1+ce^{-\Delta_{j}^{A-1}}).$
(49)

Writing $G_{\Delta_{j},A}$ in terms of its Fourier transform, we have by
truncating the exponential at $\nu=\Delta_{j}^{10A}$ (this choice will be
motivated by the estimate (55) below):

$\displaystyle G_{\Delta_{j},A}(x)$
$\displaystyle=\int_{\mathbb{R}}e^{2\pi{\rm i}\xi
x}\widehat{G}_{\Delta_{j},A}(\xi){\rm d}\xi$ (50)
$\displaystyle=\sum_{k\leq\nu}\frac{(2\pi{\rm
i}x)^{k}}{k!}\int_{\mathbb{R}}\xi^{k}\widehat{G}_{\Delta_{j},A}(\xi){\rm
d}\xi+\operatorname{O}^{*}\Big{(}\frac{(2\pi)^{\nu}x^{\nu}}{\nu!}\int_{\mathbb{R}}\xi^{\nu}\widehat{G}_{\Delta_{j},A}(\xi){\rm
d}\xi\Big{)},$

where $\operatorname{O}^{*}$ means that implicit constant is smaller than $1$
in absolute value. The polynomial term in (50) is our definition of the
polynomial $\mathcal{D}_{\Delta_{j},A}(x)$ in (48). Since the $Y_{j}$ is a sum
over primes in $(\exp(e^{t_{j-1}}),\exp(e^{t_{j}})]$, it is clear that
$\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})$ is a Dirichlet polynomial involving
integers with prime factors in that interval and that its length is at most
$\exp(2e^{t_{j}}\Delta_{j}^{10A})$. (The factor $2$ comes from the fact that
$Y_{j}$ includes squares of primes.) It remains to estimate the error term.
Since Equation (48) is trivial if $|Y_{j}-u_{j}|>\Delta_{j}^{-1}$, we assume
without loss of generality that $|Y_{j}-u_{j}|\leq\Delta_{j}^{-1}$. Therefore
the error term is

$\displaystyle\frac{(2\pi)^{\nu}}{\nu!}\int_{\mathbb{R}}\xi^{\nu}\widehat{G}_{\Delta_{j},A}(\xi){\rm
d}\xi$
$\displaystyle\leq\frac{(2\pi)^{\nu}}{\nu!}\int_{\mathbb{R}}|\xi|^{\nu}|\widehat{G}_{\Delta_{j},A}(\xi)|{\rm
d}\xi$ $\displaystyle\leq\frac{(2\pi)^{\nu}}{\nu!}\cdot
2\Delta_{j}^{2A(\nu+1)}\leq\frac{(100)^{\nu}}{\nu^{\nu}}\Delta_{j}^{3A\nu},$
(51)

where we use properties (1) and (5) above. This is $e^{-\Delta^{4A}}$ for the
choice $\nu=\Delta_{j}^{10A}$. Putting this back in (49) yields

$\mathbf{1}(Y_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}])\leq|\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})+\operatorname{O}^{*}(e^{-\Delta_{j}^{4A}})|^{2}\cdot(1+ce^{-\Delta_{j}^{A-1}}).$

The term $\operatorname{O}^{*}(e^{-\Delta_{j}^{4A}})$ can be absorbed in the
multiplicative error by adjusting $c$. The choice $A\geq 10$ ensures a decay
much better than Gaussian. ∎

It follows from Equation (47) and Lemma 2.8 that

$\mathbf{1}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\})\leq\sum_{\mathbf{u}\in\mathcal{I}}\prod_{j}|\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}(1+ce^{-\Delta_{j}^{A-1}}).$
(52)

We choose $A=20$ for the rest of the proof. The product over $j$ of
$|\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}$ is a Dirichlet polynomial of
length at most

$\exp\big{(}2\sum_{j=1}^{\ell}e^{t_{j}}\Delta_{j}^{10A}\big{)}\leq\exp\Big{(}2e^{t_{\ell}}\Delta_{\ell}^{10A}\sum_{j=1}^{\ell}\Big{(}\frac{\log_{\ell-1}t}{\log_{j-1}t}\Big{)}^{\mathfrak{s}-10A}\Big{)}\leq\exp(2e^{t_{\ell}}\Delta_{\ell}^{10A}),$

since $\mathfrak{s}\geq 10^{6}>10A$ by the choice of $\mathfrak{s}$ in (12)
and the choice $A=20$. The mean-value theorem for Dirichlet polynomials, see
Lemma A.3 (which applies by the assumption on $\ell$), implies

$\mathbf{E}\Big{[}\prod_{j}|\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}\Big{]}=(1+{\rm
o}(1))\mathbf{E}\Big{[}\prod_{j}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\Big{]},$
(53)

where $(\mathcal{Y}_{j},j\leq\ell)$ are independent random variables of the
form

$\mathcal{Y}_{j}=\sum_{e^{t_{j-1}}<\log p\leq
e^{t_{j}}}\frac{\cos\theta_{p}}{p^{1/2}}+\frac{\cos^{2}\theta_{p}}{2p},$ (54)

and $(\theta_{p},p\text{ primes})$ are independent random variables uniform on
$[0,2\pi]$. It remains to estimate
$\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\Big{]}$
for each $j$.

######  Lemma 2.9.

With the above notation, we have for $j\leq\ell$ and an absolute constant
$c>0$,

$\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\Big{]}\leq\mathbf{P}(\mathcal{Y}_{j}-u_{j}\in[-\Delta_{j}^{-A/2},\Delta_{j}^{-1}+\Delta_{j}^{-A/2}])+ce^{-\Delta_{j}^{A-1}}.$

###### Proof.

The idea is to use the approximation with $G_{\Delta_{j},A}$ in reverse. For
this, it is necessary to re-introduce the error term in Equation (50),
assuming it is small enough. On the event
$|\mathcal{Y}_{j}-u_{j}|\leq\Delta_{j}^{6A}$, the estimate (51) becomes

$\displaystyle\frac{(2\pi)^{\nu}\Delta_{j}^{6A\nu}}{\nu!}\int_{\mathbb{R}}\xi^{\nu}\widehat{G}_{\Delta_{j},A}(\xi){\rm
d}\xi$
$\displaystyle\leq\frac{(2\pi)^{\nu}}{\nu!}\cdot\Delta_{j}^{2A(4\nu+1)}\leq\frac{(100)^{\nu}}{\nu^{\nu}}\Delta_{j}^{9A\nu}.$
(55)

This is $e^{-\Delta_{j}^{4A}}$ for the choice $\nu=\Delta_{j}^{10A}$. On the
event $|\mathcal{Y}_{j}-u_{j}|>\Delta_{j}^{6A}$, Cauchy-Schwarz inequality
yields

$\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\mathbf{1}(|\mathcal{Y}_{j}-u_{j}|>\Delta_{j}^{6A})\Big{]}\leq\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{4}\Big{]}^{1/2}\cdot\mathbf{P}\big{(}|\mathcal{Y}_{j}-u_{j}|>\Delta_{j}^{6A})\big{)}^{1/2}.$

The fourth moment of
$\mathbf{E}[|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}(h)-u_{j})|^{4}]$ is
bounded by

$\mathbf{E}\Big{[}\Big{(}\sum_{\ell\leq\Delta_{j}^{10A}}\frac{(2\pi)^{\ell}}{\ell!}2\Delta_{j}^{2A(\ell+1)}(|\mathcal{Y}_{j}|+|u_{j}|)^{\ell}\Big{)}^{4}\Big{]}\ll\Delta_{j}^{2A}\,\mathbf{E}[\exp(9\pi\Delta_{j}^{2A}(|\mathcal{Y}_{j}|+4\Delta_{j}))]\ll
e^{\Delta_{j}^{5A}},$ (56)

where we used Equation (45) and the fact that
$\mathbf{E}[e^{\lambda\mathcal{Y}_{j}}]\ll\exp(\lambda^{2}\Delta_{j})$ by
Lemma A.5. The probability is bounded by Chernoff’s inequality using the same
lemma

$\mathbf{P}\big{(}|\mathcal{Y}_{j}-u_{j}|>\Delta_{j}^{6A}\big{)}\ll\exp(-\frac{1}{2}\Delta_{j}^{6A}).$
(57)

Equations (56) and (57) together imply

$\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\mathbf{1}(|\mathcal{Y}_{j}-u_{j}|>\Delta_{j}^{6A})\Big{]}\leq
e^{-\frac{1}{8}\Delta_{j}^{6A}}.$

Altogether, we have shown

$\mathbf{E}\Big{[}|\mathcal{D}_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})|^{2}\Big{]}\leq\mathbf{E}[|G_{\Delta_{j},A}(\mathcal{Y}_{j}-u_{j})+\operatorname{O}(e^{-\Delta_{j}^{4A}})|^{2}]+e^{-\frac{1}{8}\Delta_{j}^{6A}}.$

Since $G_{\Delta_{j},A}$ is in $[0,1]$ the error inside the expectation can be
made additive. The statement of the lemma then follows from property (4) of
the function $G_{\Delta_{j},A}$. ∎

The proof of Lemma 2.6 can now be concluded.

###### Proof of Lemma 2.6.

Let’s first notice that by a direct application of Berry-Esseen theorem, see
Lemma A.6, we have for any $j\geq 2$

$\mathbf{P}(\mathcal{Y}_{j}-u_{j}\in[-\Delta_{j}^{-A/2},\Delta_{j}^{-1}+\Delta_{j}^{-A/2}])=\mathbf{P}(\mathcal{N}_{j}-u_{j}\in[-\Delta_{j}^{-A/2},\Delta_{j}^{-1}+\Delta_{j}^{-A/2}])+\operatorname{O}(e^{-ce^{t_{j-1}}}).$
(58)

where $\mathcal{N}_{j}$ is a Gaussian random variable of mean $0$ with
variance $\frac{1}{2}(t_{j}-t_{j-1})+{\rm o}(1)$. For $j=1$, we use the less
accurate estimate in Lemma A.7:

$\mathbf{P}(\mathcal{Y}_{1}-u_{1}\in[-\Delta_{1}^{-A/2},\Delta_{1}^{-1}+\Delta_{1}^{-A/2}])\ll\mathbf{P}(\mathcal{N}_{1}-u_{1}\in[-\Delta_{1}^{-A/2},\Delta_{1}^{-1}+\Delta_{1}^{-A/2}]).$

Since $|u_{j}|<4\Delta_{j}$ by Equation (45), we have that
$u_{j}\cdot\Delta_{j}^{-A/2}$ is very small, and therefore by using a Gaussian
estimate, we get for all $j\geq 2$,

$\mathbf{P}(\mathcal{N}_{j}-u_{j}\in[-\Delta_{j}^{-A/2},\Delta_{j}^{-1}+\Delta_{j}^{-A/2}])=\mathbf{P}(\mathcal{N}_{j}-u_{j}\in[0,\Delta_{j}^{-1}])(1+\operatorname{O}(\Delta_{j})^{-A/4}).$

For $j=1$, the corresponding estimate holds with $\ll$ instead of $=$. We also
notice that the error term in (58) is much smaller than the probability and
can be absorbed in the multiplicative error above. Therefore we have shown for
$j\geq 2$ that

$\mathbf{P}(\mathcal{Y}_{j}-u_{j}\in[-\Delta_{j}^{-A/2},\Delta_{j}^{-1}+\Delta_{j}^{-A/2}])=\mathbf{P}(\mathcal{N}_{j}-u_{j}\in[0,\Delta_{j}^{-1}])(1+\operatorname{O}(\Delta_{j})^{-A/4}),$

and for $j=1$

$\mathbf{P}(\mathcal{Y}_{1}-u_{1}\in[-\Delta_{1}^{-A/2},\Delta_{1}^{-1}+\Delta_{1}^{-A/2}])\ll\mathbf{P}(\mathcal{N}_{1}-u_{1}\in[0,\Delta_{1}^{-1}])(1+\operatorname{O}(\Delta_{1})^{-A/4}).$

Putting this estimate back in Equation (47) using Lemmas 2.8 and 2.9 (noticing
again that the additive error in Lemma 2.9 can be made multiplicative), it
follows that

$\mathbf{P}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\})\ll\sum_{\mathbf{u}\in\mathcal{I}}\prod_{j=1}^{\ell}\mathbf{P}(\mathcal{N}_{j}\in[u_{j},u_{j}+\Delta_{j}^{-1}])(1+\operatorname{O}(\Delta_{j})^{-A/4}).$

It remains to re-express the events in terms of the partial sums of
$\mathcal{N}_{j}$, exactly as we did in Equation (46) but in reverse. By the
definition of $\mathcal{I}$ and the summability of $\Delta_{j}^{-1}$, we
conclude that

$\mathbf{P}(B_{\ell}\cap
C_{\ell}\cap\\{S_{t_{\ell}}\in(w,w+1]\\})\ll\mathbf{P}\big{(}\sum_{j=1}^{\ell}\mathcal{N}_{j}\in(w-1,w+2]\big{)}.$

Here, we dropped the intermediate restrictions on the partial sums that are no
longer needed. The right side is
$\ll\frac{1}{\sqrt{t_{\ell}}}e^{-w^{2}/t_{\ell}}$ as claimed.

∎

###  2.5 Proof of Lemma 2.4 and Lemma 2.7

We prove Equation (37). The proof of Lemma 2.4 and of Equation (36) are
similar and simpler. The proof follows closely the one of Lemma 2.6 with an
additional tool from [ABR20]. Given $\ell\geq 1$, a Dirichlet polynomial
$\mathcal{Q}$ is said to be degree-$e^{t_{\ell}}$ well-factorable if it can be
expressed as

$\prod_{1\leq\lambda\leq\ell}\mathcal{Q}_{\lambda}(s),\quad\text{ where
}\quad\mathcal{Q}_{\lambda}(s)=\sum_{\begin{subarray}{c}p|m\implies\log
p\in(e^{t_{\lambda-1}},e^{t_{\lambda}}]\\\ \Omega_{\lambda}(m)\leq
10(t_{\lambda}-t_{\lambda-1})^{10^{4}}\end{subarray}}\frac{\gamma(m)}{m^{s}},$

and $\gamma$ are arbitrary coefficients such that
$|\gamma(m)|\leq\exp(\tfrac{1}{500}e^{t})$ for every $m\geq 1$. We need the
following twisted fourth moment estimate.

######  Lemma 2.10 (Lemma 9 in [ABR20]).

Let $\ell\geq 0$ be such that
$\exp(10^{6}(t_{\ell+1}-t_{\ell})^{10^{5}}e^{t_{\ell+1}})\leq\exp(\tfrac{1}{100}e^{t})$.
Let $\mathcal{Q}$ be a degree-$e^{t_{\ell+1}}$ well-factorable Dirichlet
polynomial. Then, we have

$\mathbb{E}\Big{[}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|^{4}\cdot|\mathcal{Q}|^{2}\Big{]}\ll
e^{4(t-t_{\ell+1})}\,\mathbb{E}\Big{[}|\mathcal{Q}|^{2}\Big{]}.$

###### Proof of Lemma 2.7.

We proceed as in the proof of Lemma 2.6 by approximating the indicator
function by a Dirichlet polynomial. More precisely, using Equations (47) and
(48), the left-hand side of (37) becomes

$\ll\sum_{\mathbf{u}\in\mathcal{I}}\mathbb{E}\Big{[}|\zeta\mathcal{M}_{1}\cdots\mathcal{M}_{\ell+1}|^{4}|\mathcal{Q}_{\ell}|^{2}\prod_{j}\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}\Big{]}.$

We choose $A=20$. The polynomial
$Q=Q_{\ell}\prod_{j}\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})$ is well-
factorable, and $\mathcal{Q}_{\ell}$ is as defined in the statement of Lemma
2.7. Since the coefficients of $\mathcal{D}_{\Delta_{j},A}$ are bounded by
$\Delta_{j}^{2A(\nu+1)}$, the coefficients of $\mathcal{Q}$ are bounded by
$\exp(\tfrac{1}{500}e^{t})$. Moreover, its length is

$\leq\exp(10e^{t_{\ell+1}}(t_{\ell+1}-t_{\ell})^{10^{4}})\cdot\exp(2e^{t_{\ell}}\Delta_{\ell}^{200})<\exp(e^{t}/100),$

since $\mathfrak{s}\geq 10^{6}$ and by the assumption on $\ell$. This implies
by Lemma 2.10 that the above is

$\ll
e^{4(t-t_{\ell+1})}\sum_{\mathbf{u}\in\mathcal{I}}\mathbb{E}\Big{[}|\mathcal{Q}_{\ell}|^{2}\prod_{j}\mathcal{D}_{\Delta_{j},A}(Y_{j}-u_{j})|^{2}\Big{]}.$

The expectation splits by Lemma A.4. It remains to proceed as before from
Equation (53) to get Equation (37). ∎

##  3 Proofs of Corollaries

###  3.1 Proof of Corollary 1.2

Consider the CDF of the random variable $\log|\zeta(1/2+{\rm i}\tau)|$, i.e.,
$F(V)=\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|\leq V)$. Write for short

$S(V)=\mathbf{P}(\log|\zeta(1/2+{\rm i}\tau)|>V).$

Recall that $\tau$ is distributed uniformly on $[T,2T]$, and we write
$t=\log\log T$. Clearly, the moments (cf. Equation (5)) can be written as

$M_{\beta}=\int_{-\infty}^{+\infty}e^{\beta V}{\rm d}F(V).$

Integration by parts yields

$M_{\beta}=-e^{\beta
V}S(V)\Big{|}_{-\infty}^{+\infty}+\int_{-\infty}^{+\infty}\beta e^{\beta
V}S(V){\rm d}V.$ (59)

Since $S(V)$ is bounded by one, the boundary term at $-\infty$ is zero.
Moreover, Markov’s inequality with the fourth moment of zeta [Theorem B
[Ing26]] gives

$S(V)\leq\frac{1}{2\pi^{2}}e^{4t}e^{-4V}.$ (60)

In particular, this implies that the boundary term at $+\infty$ is zero for
$\beta<4$. The contribution to negative $V$’s in the integral in Equation (59)
is also negligible since

$\int_{-\infty}^{0}\beta e^{\beta V}S(V){\rm d}V\leq\int_{-\infty}^{0}\beta
e^{\beta V}{\rm d}V=1.$ (61)

It remains to estimate $\int_{0}^{+\infty}\beta e^{\beta V}S(V){\rm d}V$.
Consider $\beta_{-}$ and $\beta_{+}$ such that
$0<\beta_{-}<\beta<\beta_{+}<4$. These have to be chosen close enough to $0$
and to $4$ respectively. It turns out that the choices

$\displaystyle\beta_{-}$ $\displaystyle=\frac{\beta}{4}$
$\displaystyle\beta_{+}$
$\displaystyle=\beta+\frac{3}{4}(4-\beta)=3+\frac{\beta}{4}$

are adequate. The dominant contribution to the $\beta$-moment comes from the
interval $[\frac{\beta_{-}}{2}t,\frac{\beta_{+}}{2}t]$. Indeed, by Theorem
1.1, we have

$\int_{\frac{\beta_{-}}{2}t}^{\frac{\beta_{+}}{2}t}e^{\beta V}S(V){\rm
d}V\ll\int_{\frac{\beta_{-}}{2}t}^{\frac{\beta_{+}}{2}t}e^{\beta
V}\frac{e^{-V^{2}/t}}{\sqrt{t}}{\rm
d}V=e^{\frac{\beta^{2}}{4}t}\int_{\frac{\beta_{-}}{2}t}^{\frac{\beta_{+}}{2}t}\frac{e^{-(\frac{\beta}{2}t-V)^{2}/t}}{\sqrt{t}}{\rm
d}V\ll e^{\frac{\beta^{2}}{4}t}.$

The contribution of the intervals $[0,\frac{\beta_{-}}{2}t]$ is less since it
is smaller than

$\int_{0}^{\frac{\beta_{-}}{2}t}\beta e^{\beta V}{\rm d}V\leq
e^{\frac{\beta^{2}t}{8}},$ (62)

by the choice of $\beta_{-}$. For the interval
$[\frac{\beta_{+}}{2}t,\infty]$, we use the bound (60) to get that the
contribution is

$\leq e^{4t}\int_{\frac{\beta_{+}}{2}t}^{\infty}e^{(\beta-4)V}{\rm
d}V\leq\frac{1}{4-\beta}e^{t(\frac{\beta\beta_{+}}{2}-2\beta_{+}+4)}.$

This is $\ll e^{\beta^{2}t/4}$ by the choice of $\beta_{+}$.

###  3.2 Proof of Corollary 1.3

We will require the following discretization result of [FGH07]. Effectively,
this shows that the maximum of concern in Corollary 1.3 can be restricted to
those $h$ lying $1/\log T$ apart. Corollary 1.3 may also be deduced from a
more general discretization result of [AOR19], applicable to Dirichlet
polynomials.

######  Lemma 3.1 (Lemma 2.2 of [FGH07]).

Let $t^{*}$ be such that $|\zeta(1/2+{\rm
i}t^{*})|=\max_{t\in[T,2T]}|\zeta(1/2+{\rm i}t)|$. There is an absolute
constant $A>0$ such that if $|t-t^{*}|<A/\log T$ then $2|\zeta(1/2+{\rm
i}t)|>|\zeta(1/2+{\rm i}t^{*})|$.

Thus, as $\gamma$ ranges over a window of size $A/\log T$, the value of
$|\zeta(1/2+{\rm i}\gamma)|$ is close to the maximum within the window. Hence,
we deduce via a union bound that, for some universal positive constant $C>0$,

$\mathbf{P}\Big{(}\max_{|h|\leq\log^{\theta}T}|\zeta(1/2+{\rm i}\gamma+{\rm
i}h)|>e^{V}\Big{)}\leq e^{(1+\theta)t}\cdot\mathbf{P}\Big{(}|\zeta(1/2+{\rm
i}\gamma)|>\frac{1}{C}e^{V}\Big{)}.$ (63)

Corollary 1.3 now follows by setting
$V=\sqrt{1+\theta}t-\frac{1}{4\sqrt{1+\theta}}\log t+y$ (for $y={\rm o}(t/\log
t)$, $\theta\in[0,3)$), and applying Theorem 1.1.

###  3.3 Proof of Corollary 1.4

Case $\beta\geq 0$: We write

$\mathcal{Z}_{\beta}(\tau)=\frac{1}{2e^{\theta t}}\int_{|h|\leq e^{\theta
t}}|\zeta(1/2+{\rm i}\tau+{\rm i}h)|^{\beta}{\rm d}h,$

i.e., the left-hand side of Equation (8) normalized by $2e^{\theta t}$ and
with the identification $t=\log\log T$. The moment $\mathcal{Z}_{\beta}$ is a
random variable dependent on $\tau$. From now on, we use the probabilistic
convention and drop the dependence on $\tau$ from the notation. Consider also
the (normalized) Lebesgue measure of high points in the interval $[-e^{\theta
t},e^{\theta t}]$ around $\tau$:

$\mathcal{S}(V)=\frac{1}{2e^{\theta t}}\text{meas}\\{|h|\leq e^{\theta
t}:\log|\zeta(1/2+{\rm i}\tau+{\rm i}h)|>V\\}.$

Proceeding as in the proof of Corollary 1.2, we have by integration by parts:

$\mathcal{Z}_{\beta}=-e^{\beta
V}\mathcal{S}(V)\Big{|}_{-\infty}^{+\infty}+\beta\int_{-\infty}^{\infty}e^{\beta
V}\mathcal{S}(V){\rm d}V.$

Again, since $\mathcal{S}(V)\leq 1$ for all $V$, we have that the boundary
term at $V=-\infty$ is $0$.

For $V=+\infty$, it is necessary to restrict the estimate to a good event.
Define

$E=\left\\{\max_{|h|\leq e^{\theta t}}\log|\zeta(1/2+{\rm i}\tau+{\rm
i}h)|\leq m(t)+A\right\\},$ (64)

where

$m(t)=\sqrt{1+\theta}t-\frac{1}{4\sqrt{1+\theta}}\log
t=\frac{\beta_{c}}{2}t-\frac{1}{2\beta_{c}}\log t,$ (65)

and $\beta_{c}=2\sqrt{1+\theta}$. In view of Corollary 1.3 with the choice
$y=A$ (and since $\theta\in[0,3)$ by assumption), the probability of $E^{c}$
is

$\mathbf{P}(E^{c})\ll e^{-\beta_{c}A}.$ (66)

This handles the upper limit $V=+\infty$.

On the event $E$, there are clearly no values of $V$ beyond $m(t)+A$.
Moreover, as in the proof of Corollary 1.2, the contribution of negative
values is of order one (cf. Equation (61)). Finally, the bound (62) still
holds. The problem is therefore reduced to finding a good event on which to
bound

$\int_{\beta t/8}^{m(t)+A}e^{\beta V}\mathcal{S}(V){\rm d}V.$ (67)

The idea now is that $\mathcal{S}(V)$ should behave like $e^{-V^{2}/t-1/2\log
t}$, thanks to Theorem 1.1. In particular, as can be seen easily in the proof,
the dominant contribution to the integral should come from $V$’s around $\beta
t/2$. Hence, the specifics of the interval of integration do not matter much
as long as it contains this optimizer. The main technical difficulty in
implementing this idea is to control $\mathcal{S}(V)$ on a range of $V$
simultaneously.

Consider $(V_{j},1\leq j\leq J)$ the set of $V$’s in
$[\tfrac{\beta}{8}t,m(t)+A]\cap\sqrt{t}\mathbb{Z}$, and additionally define
$V_{0}=V_{1}-\sqrt{t}$ and $V_{J+1}=V_{J}+\sqrt{t}$. (The choice of the mesh
size $\sqrt{t}$ is informed by the typical fluctuation of $\log|\zeta|$.)
Define

$I_{j}=\int_{V_{j}}^{V_{j+1}}e^{\beta V}\mathcal{S}(V){\rm d}V,\quad 0\leq
j\leq J.$

Consider the events

$E_{j}=\left\\{I_{j}\leq a_{j}\int_{V_{j}}^{V_{j+1}}e^{\beta
V}\frac{e^{-V^{2}/t}}{\sqrt{t}}{\rm d}V\right\\},$ (68)

for a collection of $a_{j}$’s to be fixed later.

We have $\mathbf{P}(E_{j}^{c})\ll a_{j}^{-1}$, since by linearity and Theorem
1.1

$\displaystyle\mathbf{E}[I_{j}]=\int_{V_{j}}^{V_{j+1}}e^{\beta
V}\mathbf{E}[\mathcal{S}(V)]{\rm d}V$
$\displaystyle\ll\int_{V_{j}}^{V_{j+1}}e^{\beta
V}\frac{e^{-V^{2}/t}}{\sqrt{t}}{\rm d}V.$ (69)

The good event to consider is

$G=E\cap\left(\bigcap_{j}E_{j}\right),$

so that by (66)

$\mathbf{P}(G^{c})\ll\sum_{j}a_{j}^{-1}+e^{-\beta_{c}A}.$ (70)

On the event $G$, we have

$\int_{\beta t/8}^{m(t)+A}e^{\beta V}\mathcal{S}(V){\rm d}V\leq
e^{\beta^{2}t/4}\sum_{j}a_{j}\int_{V_{j}}^{V_{j+1}}\frac{e^{-(\frac{\beta}{2}t-V)^{2}/t}}{\sqrt{t}}{\rm
d}V.$ (71)

Since the quadratic form is maximized at $\beta t/2$, we pick for $a_{j}$:

$a_{j}=A\cdot\begin{cases}\big{(}\frac{\beta}{2}\sqrt{t}-\frac{V_{j}}{\sqrt{t}}\big{)}^{2}+\frac{1}{100}\
&\text{if $V_{j}>\beta t/2$},\\\
\big{(}\frac{\beta}{2}\sqrt{t}-\frac{V_{j+1}}{\sqrt{t}}\big{)}^{2}+\frac{1}{100}\
&\text{if $V_{j}\leq\beta t/2$ and $V_{j}<V_{j+1}\leq\beta t/2$}\\\
\frac{1}{100}&\text{if $V_{j}\leq\beta t/2$ and $V_{j+1}>\beta
t/2$}.\end{cases}$

(The term $1/100$ is simply there to make sure $a_{j}$ is bounded away from
$0$.) This choice ensures that $a_{j}\leq
A\big{(}\tfrac{1}{100}+(\frac{\beta}{2}\sqrt{t}-\frac{V}{\sqrt{t}})^{2}\big{)}$
for $V\in[V_{j},V_{j+1}]$.

Thus, on one hand from Equation 71, we have on $G$

$\displaystyle\int_{\beta t/8}^{m(t)+A}e^{\beta V}\mathcal{S}(V){\rm d}V$
$\displaystyle\leq Ae^{\beta^{2}t/4}\int_{\beta
t/8}^{m(t)+A}\left(\frac{1}{100}+\left(\frac{\beta}{2}\sqrt{t}-\frac{V}{\sqrt{t}}\right)^{2}\right)\cdot\frac{e^{-(\frac{\beta}{2}t-V)^{2}/t}}{\sqrt{t}}{\rm
d}V$ $\displaystyle\leq
Ae^{\beta^{2}t/4}\int_{\frac{(\beta-\beta_{c})}{2}\sqrt{t}+{\rm
o}(1)}^{\frac{3\beta}{8}\sqrt{t}}\left(\frac{1}{100}+u^{2}\right)e^{-u^{2}}{\rm
d}u$ $\displaystyle\leq Ae^{\beta^{2}t/4},$

where the last bound follows by integrating over the whole line. On the other
hand, from Equation (70) the probability of $G^{c}$ is

$\mathbf{P}(G^{c})\ll\sum_{j}a_{j}^{-1}+e^{-\beta_{c}A}\ll\frac{1}{A}.$

The $a_{j}$’s are summable since $V_{j}\in\sqrt{t}\mathbb{Z}$. This proves
Equation (8).

Case $\beta>\beta_{c}$: We can use a reduction as in the previous case. We use
the same event $E$ in (64) for the maximum. For a lower bound on the values of
$V$, we take $\beta_{c}t/4$ since

$\int_{0}^{\beta_{c}t/4}e^{\beta V}\mathcal{S}(V){\rm d}V\leq
e^{\tfrac{\beta_{c}}{4}\beta t},$

which is much smaller than the the desired bound. Therefore, it remains to
estimate

$\int_{\beta_{c}t/4}^{m(t)+A}e^{\beta V}\mathcal{S}(V){\rm d}V.$ (72)

The partitioning of the interval of integration is more delicate as it is
close to the level of the maximum. A mesh size of $1$ instead of $\sqrt{t}$ is
needed. More precisely, we take $(V_{j},1\leq j\leq J)$ to be
$[\tfrac{\beta_{c}}{4}t,m(t)+A]\cap\mathbb{Z}$. The events $E_{j}$ are defined
as in (68). As before, we take $G=E\cap(\bigcap_{j}E_{j})$. The difference
here is that the optimizer lies outside the interval, so the bound can be
sharpened. On the event $G$, the above becomes

$\leq\sum_{j}a_{j}\int_{V_{j}}^{V_{j+1}}e^{\beta
V}\frac{e^{-V^{2}/t}}{\sqrt{t}}{\rm d}V.$

The change of variable $V=m(t)+y$ yields (with $y_{j}=V_{j}-m(t)$)

$\displaystyle e^{\beta m(t)}\sum_{j}a_{j}\int_{y_{j}}^{y_{j+1}}$
$\displaystyle e^{\beta
y}\frac{e^{-m(t)^{2}/t}}{\sqrt{t}}e^{-\frac{2m(t)y}{t}}e^{-y^{2}/t}{\rm d}y$
(73) $\displaystyle\leq e^{\beta
m(t)-(1+\theta)t}\sum_{j}a_{j}\int_{y_{j}}^{y_{j+1}}e^{(\beta-\beta_{c})y}e^{y\tfrac{(\log
t)^{2}}{4\beta_{c}^{2}t}}{\rm d}y,$

since $m(t)^{2}=(1+\theta)t-\frac{1}{2}\log t+\tfrac{(\log
t)^{2}}{4\beta_{c}^{2}}$ and $e^{-y^{2}/t}\leq 1$. We pick
$a_{j}=A(1+y_{j}^{2})$ if $y_{j}$ is positive, and $a_{j}=A(1+y_{j+1}^{2})$ if
$y_{j+1}$ is negative. If $y_{j}<0<y_{j+1}$ then set $a_{j}=A$. This choice
ensures that $a_{j}\leq A(2+y^{2})$ for $y\in[y_{j},y_{j+1}]$, the term $2$
taking care of the values close to $0$.

This gives that Equation (73) is bounded by

$\displaystyle\leq Ae^{\beta
m(t)-(1+\theta)t}\int_{-\infty}^{A}(2+y^{2})e^{(\beta-\beta_{c})y}e^{y\tfrac{(\log
t)^{2}}{4\beta_{c}^{2}t}}{\rm d}y$
$\displaystyle\leq\Big{(}\tfrac{2A}{(\beta-\beta_{c})^{3}}+\tfrac{A(A^{2}+2)}{\beta-\beta_{c}}\Big{)}e^{(\beta-\beta_{c}+1)A}\cdot
e^{\beta m(t)-(1+\theta)t}$

since $e^{y\tfrac{(\log t)^{2}}{4\beta_{c}^{2}t}}\leq e^{A}$, and by direct
integration of $(2+y^{2})e^{(\beta-\beta_{c})y}$. The probability of $G^{c}$
is then

$\mathbf{P}(G^{c})=e^{-\beta_{c}A}+\sum_{j}a_{j}^{-1}\ll\frac{1}{A}.$

This proves the corollary in the case $\beta>\beta_{c}$.

######  Remark (Case $\beta=\beta_{c}$).

Since it is possible to improve the bound (8) in the range $\beta>\beta_{c}$,
one might hope to do the same at $\beta=\beta_{c}$. This is possible in the
case $\theta=0$, as discussed in the next section, but it is not expected to
be possible for $\theta>0$. Indeed, in this range of $\theta$, the above proof
should be optimal. In fact, Equation (9) would become (dropping the $a_{j}$’s
for simplicity)

$\displaystyle e^{\beta_{c}m(t)}\int_{-\infty}^{A}e^{\beta
y}\frac{e^{-m(t)^{2}/t}}{\sqrt{t}}e^{-\frac{2m(t)y}{t}}e^{-y^{2}/t}{\rm
d}t\leq e^{A}\cdot
e^{\frac{\beta_{c}^{2}}{4}t}\int_{-\infty}^{A}\frac{e^{-y^{2}/t}}{\sqrt{t}}{\rm
d}y.$ (74)

This is because $e^{-(1+\theta)t}=e^{-\frac{\beta_{c}^{2}}{4}t}$ and
$t^{-\frac{\beta}{2\beta_{c}}}=t^{-1/2}$. The integral is now finite, so one
recovers the bound (8) up to a factor of order one.

##  4 Relation to Theorem 1.5 for $\theta=0$

We briefly explain an alternative approach to proving a sharp upper bound to
the $\beta_{c}$-moment in the case $\theta=0$. It is based on the measure of
the level sets in the spirit of the proof of Corollary 1.4.

The deterministic level of the maximum is now by Equation (7)

$m(t)=t-\frac{3}{4}\log t=\frac{\beta_{c}}{2}t-\frac{3}{2\beta c}\log t.$

There is a factor $3$ in the logarithmic correction and not $1$ as in (65).
The important observation is that the typical measure of the level sets
$\mathcal{S}(m(t)+y)$ is no longer $e^{-t}e^{-2y}e^{-y^{2}/t}$ as for the case
$\theta>0$. In fact, the proof of (7) in [ABR20] also shows that

$\mathcal{S}(m(t)+y)\leq Ae^{-t}\cdot|y|e^{-2y}e^{-y^{2}/2t},\quad|y|={\rm
o}(t),$ (75)

except on an event of probability $A$. This is what is expected from the study
of the extreme values of log-correlated processes, see for example Theorem 1.1
and Lemma 4.2 in [CHL19]. We explain how the additional $y$ in the decay is
responsible for the extra $1/\sqrt{t}$ factor in the size of the moment. The
integral (72) with $\beta=\beta_{c}=2$ becomes

$\displaystyle\int_{t/2}^{m(t)+A}e^{2V}\mathcal{S}(V){\rm d}V$
$\displaystyle\leq A\frac{e^{2t}}{t^{3/2}}\int_{-t/2+\tfrac{3}{4}\log
t}^{A}e^{-t}|y|e^{-y^{2}/t}{\rm d}y$ (76)
$\displaystyle=A\frac{e^{t}}{t^{1/2}}\int_{-t/2+\tfrac{3}{4}\log
t}^{A}\frac{|y|}{\sqrt{t}}\frac{e^{-y^{2}/t}}{\sqrt{t}}{\rm d}y$
$\displaystyle=A\frac{e^{t}}{t^{1/2}}\int_{-\sqrt{t}/2+{\rm
o}(1)}^{A/\sqrt{t}}|u|e^{-u^{2}}{\rm d}u.$

The last integral is of order one. At criticality, there is now an extra
factor $1/\sqrt{t}$ coming from $t^{3/2}$ that is left, thereby giving the
overall magnitude of $\frac{e^{t}}{t^{1/2}}$ for the moment. It is also
important to observe that, because of the $\sqrt{t}$-normalization in the
integral, it is not necessary to know the level of the maximum up to order one
as in Equation (7).

##  Appendix A Appendix

The appendix gathers known results on moments of Dirichlet polynomials and
probability estimates of random models.

###  A.1 Moments of Dirichlet Polynomials

######  Lemma A.1.

Let $\widetilde{S}_{j}$ as in Equation (14). For any integers $t/2\leq j\leq
k$ and $2q\leq e^{t-k}$, we have

$\mathbb{E}[|\widetilde{S}_{k}-\widetilde{S}_{j}|^{2q}]\ll q!(k-j+1)^{q}.$

###### Proof.

This is the content of [Sou09, Lemma 3]. ∎

With the choice $q=\lceil\frac{V^{2}}{k-j+1}\rceil$, Markov’s inequality,
Lemma A.1 and Stirling’s formula imply

$\mathbb{P}\Big{(}|\widetilde{S}_{k}-\widetilde{S}_{j}|>V\Big{)}\ll\frac{V+1}{(k-j)^{1/2}}\exp\Big{(}-\frac{V^{2}}{k-j+1}\Big{)}.$
(77)

Lemma 16 of [ABR20]] gives a more precise estimate for the moments of the real
part $S_{j}$.

######  Lemma A.2.

For any integers $t/2\leq j<k$ and $2q\leq e^{t-k}$ we have

$\mathbb{E}[|S_{k}-S_{j}|^{2q}]\ll\frac{(2q)!}{2^{q}q!}\left(\frac{k-j}{2}\right)^{q}.$

Moreover, there exists $C>0$ such that for any $j<k$, and $2q\leq e^{t-k}$
such that

$\mathbb{E}[|S_{k}-S_{j}|^{2q}]\ll\sqrt{q}\frac{(2q)!}{2^{q}q!}\left(\frac{k-j+C}{2}\right)^{q}.$
(78)

As in Equation (77), one gets a Gaussian decay from Lemma A.2 for the choice
$q=\lceil\frac{V^{2}}{2(k-j+1)}\rceil$

$\mathbf{P}\left(|S_{k}-S_{j}|>V\right)\ll e^{-\frac{V^{2}}{k-j}},$ (79)

when $j>t/2$ and $V^{2}\leq\frac{k-j}{2}e^{t-k}$.

We now explain the link between Dirichlet polynomials and the random model
(53). We consider the following general setup. Let $(\theta_{p},p\text{
prime})$ be a sequence of IID random variables, uniformly distributed on
$[0,2\pi]$. For an integer $n$ with prime factorization
$n=p_{1}^{\alpha_{1}}\ldots p_{k}^{\alpha_{k}}$ with $p_{1},\ldots,p_{k}$ all
distinct, define the random variable

$Z_{n}=\prod_{j=1}^{k}\exp({\rm i}\alpha_{j}\theta_{p_{j}}).$

By construction, we have the orthogonality relation
$\mathbb{E}[Z_{n}\overline{Z}_{m}]=\mathbf{1}_{n=m}$. Therefore, for an
arbitrary sequence $a(n)$ of complex numbers, the following holds

$\sum_{n\leq N}|a(n)|^{2}=\mathbb{E}\Big{[}\Big{|}\sum_{n\leq
N}a(n)Z_{n}\Big{|}^{2}\Big{]}.$

The expectation for the random variable is directly related to the mean-value
of the square of Dirichlet polynomial, see [MV07, Corollary 3].

######  Lemma A.3.

We have

$\mathbb{E}\Big{[}\Big{|}\sum_{n\leq N}a(n)n^{{\rm
i}\tau}\Big{|}^{2}\Big{]}=\Big{(}1+\operatorname{O}\Big{(}\frac{N}{T}\Big{)}\Big{)}\sum_{n\leq
N}|a(n)|^{2}=\Big{(}1+\operatorname{O}\Big{(}\frac{N}{T}\Big{)}\Big{)}\mathbb{E}\Big{[}\Big{|}\sum_{n\leq
N}a(n)Z_{n}\Big{|}^{2}\Big{]}.$

A direct consequence of Lemma A.3 is the splitting of the expectation for
Dirichlet polynomials involving different range of primes, see for example
[ABR20, Lemma 14].

######  Lemma A.4.

Let

$A(s)=\sum_{\begin{subarray}{c}n\leq N\\\ p|n\implies p\leq
w\end{subarray}}\frac{a(n)}{n^{s}}\text{ and
}B(s)=\sum_{\begin{subarray}{c}n\leq N\\\ p|n\implies
p>w\end{subarray}}\frac{b(n)}{n^{s}}$

be two Dirichlet polynomials with $N\leq T^{1/4}$. Then, we have

$\mathbb{E}[|A(\tfrac{1}{2}+{\rm i}\tau)|^{2}\,|B(\tfrac{1}{2}+{\rm
i}\tau)|^{2}]=(1+\operatorname{O}(T^{-1/2}))\mathbb{E}[|A(\tfrac{1}{2}+{\rm
i}\tau)|^{2}]\,\mathbb{E}[|B(\tfrac{1}{2}+{\rm i}\tau)|^{2}].$

###  A.2 Estimates for the random model

Recall the definition of the random model in Equation (53).

$\mathcal{Y}_{j}=\sum_{e^{t_{j-1}}<\log p\leq
e^{t_{j}}}\frac{\cos\theta_{p}}{p^{1/2}}+\frac{\cos^{2}\theta_{p}}{2p}.$ (80)

The moment generating function is easily estimated using the independence
between the $\theta_{p}$’s.

######  Lemma A.5.

For $\lambda<\exp(\frac{1}{2}e^{t_{j}})$, we have

$\mathbf{E}[\exp(\lambda\mathcal{Y}_{j})]\ll\exp\Big{(}\frac{\lambda^{2}}{4}(t_{j}-t_{j-1})\Big{)}.$

###### Proof.

See for example [ABR20, Lemma 15]. ∎

The comparison between the random model and the Gaussian model can be made
more precise at the level of the probabilities. A version was proved in
[ABH17, Proposition 2.11] using a Berry-Esseen estimate. See also [ABR20,
Lemma 20].

######  Lemma A.6.

For $j\geq 2$, let $\mathcal{N}_{j}$ be a Gaussian random variable of mean $0$
and variance $\frac{1}{2}(t_{j}-t_{j-1})$. There exists a constant $c>0$ such
that, for any interval $A$ and $j\geq 2$,

${\mathbb{P}}\Big{(}{\mathcal{Y}}_{j}\in
A\Big{)}=\mathbb{P}\Big{(}\mathcal{N}_{j}\in
A\Big{)}+\operatorname{O}(e^{-ce^{j/2}}).$

In the case $j=1$ above, the variable $\mathcal{Y}_{1}$ is not asymptotically
Gaussian because of the small primes. Nevertheless, the following estimate
holds by a saddle-point method [ABR20, Lemma 18].

######  Lemma A.7.

Let $|v|\leq 100r$. Then, for $r>1000$ and for all $\Delta\geq 1$, we have

$\mathbb{P}(\mathcal{Y}_{1}\in[v,v+\Delta^{-1}])\asymp\frac{1}{\Delta}\cdot\frac{1}{\sqrt{r}}\exp\Big{(}-\frac{v^{2}}{r}\Big{)}.$

## References

  * [AAB+21] E. Amzallag, L.-P. Arguin, E. Bailey, K. Hui, and R. Rao.  Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem.  Experimental Mathematics, 0(0):1–13, 2021.
  * [ABB+19] L.-P. Arguin, D. Belius, P. Bourgade, M. Radziwiłł, and K. Soundararajan.  Maximum of the Riemann zeta function on a short interval of the critical line.  Comm. Pure Appl. Math., 72(3):500–535, 2019.
  * [ABH17] Louis-Pierre Arguin, David Belius, and Adam J Harper.  Maxima of a randomized Riemann zeta function, and branching random walks.  The Annals of Applied Probability, 27(1):178–215, 2017.
  * [ABR20] L.-P. Arguin, P. Bourgade, and M. Radziwiłł.  The Fyodorov-Hiary-Keating Conjecture. I.  Preprint arXiv:2007.00988, 2020.
  * [ABR22] L.-P. Arguin, P. Bourgade, and M. Radziwiłł.  The Fyodorov-Hiary-Keating Conjecture. II.  in preparation, 2022.
  * [ADH21] L.-P. Arguin, G. Dubach, and L. Hartung.  Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.  Preprint arXiv:2103.04817, 2021.
  * [AOR19] L.-P. Arguin, F. Ouimet, and M. Radziwiłł.  Moments of the Riemann zeta function on short intervals of the critical line.  Preprint arXiv:1901.04061, 2019.
  * [BH15] Anton Bovier and Lisa Hartung.  Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime.  ALEA Lat. Am. J. Probab. Math. Stat., 12(1):261–291, 2015.
  * [BK22] E. C. Bailey and J. P. Keating.  Maxima of log-correlated fields: some recent developments.  Journal of Physics A: Mathematical and Theoretical, 55(5):053001, jan 2022.
  * [BKL02] Anton Bovier, Irina Kurkova, and Matthias Löwe.  Fluctuations of the free energy in the REM and the $p$-spin SK models.  Ann. Probab., 30(2):605–651, 2002.
  * [Bov06] Anton Bovier.  Statistical mechanics of disordered systems, volume 18 of Cambridge Series in Statistical and Probabilistic Mathematics.  Cambridge University Press, Cambridge, 2006.  A mathematical perspective.
  * [Bov17] Anton Bovier.  Gaussian processes on trees: From spin glasses to branching Brownian motion, volume 163.  Cambridge University Press, 2017.
  * [CHL19] Aser Cortines, Lisa Hartung, and Oren Louidor.  The structure of extreme level sets in branching Brownian motion.  Ann. Probab., 47(4):2257–2302, 2019.
  * [DBMN19] Martina Dal Borgo, Pierre-Loïc Méliot, and Ashkan Nikeghbali.  Local limit theorems and mod-$\phi$ convergence.  ALEA Lat. Am. J. Probab. Math. Stat., 16(1):817–853, 2019.
  * [FGH07] D. W. Farmer, S. M. Gonek, and C. P. Hughes.  The maximum size of $L$-functions.  J. Reine Angew. Math., 609:215–236, 2007.
  * [FHK12] Y. V. Fyodorov, G. A Hiary, and J. P. Keating.  Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.  Physical Review Letters, 108(17):170601, 2012.
  * [FK14] Y. V. Fyodorov and J. P. Keating.  Freezing transitions and extreme values: random matrix theory, and disordered landscapes.  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2007):20120503, 2014.
  * [FMN16] V. Féray, P.-L. Méliot, and A. Nikeghbali.  Mod-$\phi$ convergence.  Springer Briefs in Probability and Mathematical Statistics. Springer, Cham, 2016.  Normality zones and precise deviations.
  * [Har13] A. J. Harper.  Sharp conditional bounds for moments of the Riemann zeta function.  Preprint arXiv:1305.4618, 2013.
  * [Har19] A. J. Harper.  On the partition function of the Riemann zeta function, and the Fyodorov–Hiary–Keating conjecture.  Preprint arXiv:1906.05783, 2019.
  * [HKO01] C. P. Hughes, J. P. Keating, and N. O’Connell.  On the characteristic polynomial of a random unitary matrix.  Comm. Math. Phys., 220(2):429–451, 2001.
  * [HL18] G. H. Hardy and J. E. Littlewood.  Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes.  Acta Mathematica, 41:119–196, 1918.
  * [HRS19] W. Heap, M. Radziwiłł, and K. Soundararajan.  Sharp upper bounds for fractional moments of the Riemann zeta function.  Q. J. Math., 70(4):1387–1396, 2019.
  * [HS20] Winston Heap and Kannan Soundararajan.  Lower bounds for moments of zeta and $l$-functions revisited.  arXiv preprint arXiv:2007.13154, 2020.
  * [Ing26] A. E. Ingham.  Mean-value theorems in the theory of the Riemann zeta-function.  Proceedings of the London Mathematical Society, 2(1):273–300, 1926\.
  * [Ino19] Shōta Inoue.  On the logarithm of the Riemann zeta-function and its iterated integrals.  arXiv preprint arXiv:1909.03643, 2019.
  * [KS00] J. P. Keating and N. C. Snaith.  Random matrix theory and $\zeta(1/2+it)$.  Communications in Mathematical Physics, 214(1):57–89, 2000.
  * [MV07] Hugh L. Montgomery and Robert C. Vaughan.  Multiplicative number theory. I. Classical theory, volume 97 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2007.
  * [Naj18] J. Najnudel.  On the extreme values of the Riemann zeta function on random intervals of the critical line.  Probab. Theory Related Fields, 172(1-2):387–452, 2018.
  * [Pow18] Ellen Powell.  Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation.  Electronic Journal of Probability, 23(none):1 – 26, 2018.
  * [Rad11] M. Radziwiłł.  Large deviations in Selberg’s central limit theorem.  Preprint arXiv:1108.5092, 2011.
  * [RS17] Maksym Radziwiłł and Kannan Soundararajan.  Selberg’s central limit theorem for $\log{|\zeta(1/2+it)|}$.  Enseign. Math., 63(1-2):1–19, 2017.
  * [Sel46] Atle Selberg.  Contributions to the theory of the Riemann zeta-function.  Arch. Math. Naturvid., 48(5):89–155, 1946.
  * [Sel92] Atle Selberg.  Old and new conjectures and results about a class of Dirichlet series.  In Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), pages 367–385. Univ. Salerno, Salerno, 1992\.
  * [Sou09] K. Soundararajan.  Moments of the Riemann zeta function.  Ann. of Math. (2), 170(2):981–993, 2009.
  * [Sou21] K. Soundararajan.  The distribution of values of zeta and $L$-functions.  arXiv:2112.03389, 2021.