1. Introduction
Let
$(X(k))_{k\geqslant 1}$
be a sequence of independent standard complex Gaussian random variables, where the real and imaginary parts of X(k) are independently distributed like real Gaussian random variables with mean 0 and variance
${1}/{2}$
. Consider a sequence of random variables
$(A(n))_{n\geqslant 0}$
defined by the formal power series identity
\begin{equation} \exp\bigg( \sum_{k=1}^{+\infty} \frac{X(k)}{\sqrt{k}}z^k \bigg)= \sum_{n=0}^{+\infty} A(n)z^n.\end{equation}
Let
$\mathcal{P}$
be the set of the prime numbers, a Steinhaus random multiplicative function is obtained by letting
$(f(p))_{p\in \mathcal{P}}$
be a sequence of independent Steinhaus random variables (i.e distributed uniformly on the unit circle
$\{ |z|=1\}$
), and then setting
\begin{align*} f(n)\,:\!= \prod_{p^{a}||n} f(p)^{a} \text{ for all } n\in \mathbb{N},\end{align*}
where
$ p^{a}||n$
means that
$p^{a}$
is the highest power of p that divides n. In two recent papers ([Reference Gerspach2, Reference Soundararajan and Zaman7]), the sequence of random variables
$(A(n))_{n\geqslant 0}$
has been interpreted as an analogue to the Steinhaus random multiplicative function.
Recently, there has been much focus regarding the moments and almost sure bounds for the mean values of random multiplicative functions. For the lower bound, Harper [Reference Harper5] proved, using a Multiplicative Chaos techniques, that for any function V(x) tending to infinity with x, there almost surely exists arbitrarily large values of x for which
\begin{equation} \big|M_f(x) \big| \gg \frac{\sqrt{x} (\log_2 x)^{1/4}}{V(x)},\end{equation}
where
$M_f(x)\,:\!=\sum_{n\leqslant x}f(n)$
. Here and in the sequel
$\log_k$
denotes the k-fold iterated logarithm. In [Reference Harper4], Harper proved when
$x \to +\infty$
\begin{equation} \mathbb{E}\bigg[ \big| M_f(x)\big| \bigg] \asymp \frac{\sqrt{x}}{(\log_2 x)^{1/4}}.\end{equation}
This discrepancy of a factor
$\sqrt{\log_2 x}$
between the first moment and the almost sure behaviour is similar to the Law Iterated Logarithm for independent random variables. For this reason Harper conjectured that for any fixed
$\varepsilon \gt 0$
, we might have almost surely, as
$x \to +\infty$
\begin{equation} M_f(x)\ll \sqrt{x} (\log_2 x)^{1/4+\varepsilon}\end{equation}
(see the [Reference Harper5, introduction] for more details). In [Reference Caich1], the author proved
\begin{equation} M_f(x)\ll \sqrt{x} (\log_2 x)^{3/4+\varepsilon}.\end{equation}
In [Reference Soundararajan and Zaman7], Soundararajan and Zaman were motivated by the outcome and examined the moments of A(n), revealing that they resemble those in the random multiplicative functions. They proved the analogue of (1·3), that we have
\begin{align*}\mathbb{E}[|A(n)|] \asymp \frac{1}{(\log n)^{1/4}}.\end{align*}
More recently, in [Reference Gerspach2], Gerspach, proved the analogue of (1·2) that for any function V(n) tending to infinity with n, there almost surely exist arbitrarily large values of n for which
\begin{align*}|A(n)| \geqslant \frac{(\log n)^{1/4}}{V(n)}.\end{align*}
The main aim of this paper is to prove the analogue of the almost sure inequality (1·5).
Theorem 1·1. Let
$\varepsilon \gt 0$
. Let
$(A(n))_{n\geqslant 0}$
as defined in (1·1). We have almost surely, as n tends infinity
\begin{equation} A(n) \ll (\log n)^{\frac{3}{4}+ \varepsilon}.\end{equation}
We aim to enhance and simplify the demonstration of theorem 1·1 given in [Reference Caich1], in the case for the Steinhaus and Rademacher multiplicative function. Our objective is to create a gentle proof for those seeking a comprehensible grasp of the theorem in the context of this model.
2. Preliminaries
2·1. Notation
Let’s start with some definitions. Let
$(\Omega, \mathcal{F}, \mathbb{P})$
be a probability space. We call a filtration every sequence
$(\mathcal{F}_n)_{n \geqslant 1}$
of increasing sub-
$\sigma$
-algebras of
$\mathcal{F}$
. We say that a sequence of real random variables
$(Z_n)_{n \geqslant 1}$
is a submartingale (resp. supermartingale) sequence with respect to the filtration
$(\mathcal{F}_n)_{n \geqslant 1}$
, if the following properties are satisfied:
(i)
$Z_n$
is
$\mathcal{F}_n$
measurable;
(ii)
$\mathbb{E}[|Z_n|]\lt +\infty$
;
(iii)
$\mathbb{E}[Z_{n+1} \, | \, \mathcal{F}_n] \geqslant Z_n$
(resp.
$\mathbb{E}[Z_{n+1}\, | \,\mathcal{F}_n] \leqslant Z_n$
) almost surely.
We say that
$(Z_n)_{n \geqslant 1}$
is a martingale difference sequence with respect to the same filtration
$(\mathcal{F}_n)_{n \geqslant 1}$
if:
(iv)
$Z_n$
is
$\mathcal{F}_n$
measurable;
(v)
$\mathbb{E}[|Z_n|]\lt +\infty$
;
(vi)
$\mathbb{E}[Z_{n+1} \, | \,\mathcal{F}_n] =0$
almost surely.
An event
$ E \in \mathcal{F}$
happens almost surely if
$\mathbb{P}[E]=1$
.
Let Z be a random variable and let
$\mathcal{H}_1\subset \mathcal{H}_2 \subset \mathcal{F}$
some sub-
$\sigma$
-algebras, we have
\begin{align*}\mathbb{E}\big[ \mathbb{E}\big[ Z \, \big| \,\mathcal{H}_2 \big] \,\big| \, \mathcal{H}_1 \big]=\mathbb{E}\big[ Z\, \big| \, \mathcal{H}_1 \big].\end{align*}
2·2. Some properties
We follow the notation of Soundararajan and Zaman in [Reference Soundararajan and Zaman7]. Let
$(X(k))_{k\geqslant 1}$
be a sequence of independent standard complex Gaussian random variables. By a partition
$\lambda $
we mean a non-increasing sequence of non-negative integers
$\lambda_1 \geqslant \lambda_2 \geqslant ...,$
with
$\lambda_n =0$
from a certain point onwards. We denote by
$|\lambda| = \lambda_1 + \lambda_2 + \lambda_3 ...,$
and for each integer
$k \geqslant 1$
we denote by
$m_k=m_k(\lambda)$
the number of parts in
$\lambda$
that equal to k. With this notation, let
\begin{equation} a(\lambda)\,:\!= \prod_{k\geqslant 0} \bigg( \frac{X(k)}{\sqrt{k}}\bigg)^{m_k} \frac{1}{m_k! },\end{equation}
we then have
\begin{align*}\exp\bigg( \sum_{k=1}^{+\infty} \frac{X(k)}{\sqrt{k}}z^k \bigg)= \sum_{\lambda} a(\lambda) z^{|\lambda|},\end{align*}
thus, for every
$n\geqslant 0$
\begin{align*}A(n)=\sum_{|\lambda|=n} a(\lambda).\end{align*}
Note that for a standard complex Gaussian Z, we have
\begin{align*}\mathbb{E}\big[Z^n \overline{Z}^m\big]=\left\{ \begin{array}{@{}ll@{}} n! & \text{if}\ n =m, \\ 0 & \text{otherwise.} \end{array}\right.\end{align*}
It follows that if
$\lambda \neq \lambda^{\prime}$
\begin{align*}\mathbb{E}\big[a(\lambda)\overline{a(\lambda^{\prime})} \big] =0.\end{align*}
If
$\lambda=\lambda^{\prime}$
then
\begin{align*}\mathbb{E}\big[ |a(\lambda)|^2\big]=\prod_{k\geqslant 0} \frac{1}{k^{m_k}m_k!^2} \mathbb{E}\big[|X(k)|^{2m_k}\big]= \prod_{k\geqslant 0} \frac{1}{k^{m_k}m_k!}.\end{align*}
Thus
\begin{align*}\mathbb{E}\big[ |A(n)|^2\big] = \sum_{|\lambda|=n} \mathbb{E}\big[|a(\lambda)|^2\big] =\sum_{|\lambda|=n} \prod_{k\geqslant 0} \frac{1}{k^{m_k}m_k!} =1.\end{align*}
The final step is derived from the well-known formula for calculating the number of permutations in the symmetric group
$S_n$
whose cycle decomposition corresponds to the partition
$\lambda$
.
One can study A(n) through its generating function. Note that by Cauchy’s Theorem, for
$n\leqslant R$
, we have
\begin{align*}A(n)= \frac{1}{2 \pi i}\int_{|z|=1} F_R(z)\frac{{\rm d}z}{z^{n+1}},\end{align*}
where
\begin{equation} F_R(z)\,:\!= \exp\bigg(\sum_{k\leqslant R} \frac{X(k)}{\sqrt{k}}z^k\bigg).\end{equation}
We start our proof by stating some tools.
Lemma 2·1. (Borel–Cantelli’s First Lemma). Let
$(A_n)_{n \geqslant 1}$
be sequence of events. Assuming that
$\sum_{n=1}^{+ \infty}\mathbb{P}[A_n] \lt + \infty $
then
$\mathbb{P}[\limsup_{n \to + \infty} A_n]=0$
.
Proof. See [Reference Gut3, theorem 18·1].
Lemma 2·2. Let
$Z=(Z_n)_{1 \leqslant n \leqslant N}$
be a complex martingale difference sequence with respect to a filtration
$\mathcal{F}=(\mathcal{F}_n)_{1 \leqslant n \leqslant N }$
. We assume that for each n,
$Z_n$
is bounded almost surely. Furthermore, assume that we have
$|Z_n| \leqslant S_n$
almost surely, where
$(S_n)_{1 \leqslant n \leqslant N }$
is a real predictable process with respect to the same filtration. We set the event
$\Sigma\,:\!= \bigg\{ \sum_{1\leqslant n \leqslant N}S_n^2 \leqslant T\bigg\}$
, where T is a deterministic constant. Then, for any
$\varepsilon \gt 0$
,
\begin{align*} \mathbb{P}\bigg[\bigg\{\bigg|\sum_{1\leqslant n \leqslant N}Z_n\bigg|\geqslant \varepsilon\bigg\}\, \bigcap \, \Sigma \bigg] \leqslant 2\exp\bigg(\frac{-\varepsilon^2}{10T}\bigg).\end{align*}
Proof. See [Reference Caich1, lemma 3·13].
Lemma 2·3. Let
$R \geqslant 1$
be a real number and
$F_R(z)$
as in (2·2). Uniformly for
$1/2 \leqslant q \leqslant 1$
and
$1\leqslant r \leqslant {\rm e}^{1/R}$
, we have
\begin{align*} \mathbb{E}\bigg[\bigg( \frac{1}{2\pi} \int_{0}^{2\pi} |F_R(r{\rm e}^{i\theta})|^2{\rm d}\theta \bigg)^q\bigg]\ll \bigg(\frac{R}{1+(1-q)\sqrt{\log R}}\bigg)^q. \end{align*}
Proof. See [Reference Soundararajan and Zaman7, proposition 3·2].
3. Reduction of the problem
The aim of this section is to reduce the problem to something simple to deal with. We want to prove that the event
\begin{align*} \mathcal{A}\,:\!=\big\{ |A(n)|\gt 4(\log n )^{3/4 + \varepsilon} \text{, for infinitely many }n\big\}\end{align*}
holds with null probability. We adopt the reasoning from [Reference Caich1] and set
$X_{\ell}$
to be equal to
$2^{\ell^K}$
, where
$K={25}/{\varepsilon}$
. It suffices to prove that the event
\begin{align*} \mathcal{B}\,:\!=\bigg\{ \sup_{X_{\ell -1}\lt n\leqslant X_{\ell}}\frac{|A(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 4 \text{, for infinitely many }\ell\bigg\}\end{align*}
holds with null probability. We set
\begin{align*}\mathcal{B}_{\ell} \,:\!= \bigg\{ \sup_{X_{\ell -1}\lt n\leqslant X_{\ell}}\frac{|A(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 4 \bigg\}.\end{align*}
In order to prove Theorem 1·1 using Borel–Cantelli’s First Lemma 2·1, it is enough to establish the convergence of the sum
$ \sum_{\ell \geqslant 1} \mathbb{P}\big[\mathcal{B}_{\ell}\big]$
.
Arguing as Lau–Tenenbaum–Wu in [Reference Lau, Tenenbaum and Wu6, proof of lemma 3·1] and recently in [Reference Caich1, start of section 5·1], we consider for every
$j\geqslant 0 $
\begin{equation}y_j=\bigg\lfloor\frac{2^{\ell^K}{\rm e}^{j/\ell}}{2^{K\ell^{K-1}}}\bigg\rfloor \text{ and } \widetilde{y}_j\,:\!= \frac{2^{\ell^K}{\rm e}^{j/\ell}}{2^{K\ell^{K-1}}}.\end{equation}
Let J be minimal under the constraint
$y_J \geqslant X_{\ell}$
which means
\begin{equation} J_{\ell}=J\,:\!=\lceil K \ell^{K} \log 2 \rceil \ll \ \ell^{K}.\end{equation}
Note that
$\ell^{K} ={1}/{\log 2} \log X_{\ell} \asymp \log n \asymp \log y_j $
for any
$n \in ]X_{\ell -1},X_{\ell}]$
and
$~1\leqslant~ j\leqslant ~J$
.
Let
$n \in ]X_{\ell -1},X_{\ell}]$
, we start by splitting A(n) according to the size of
$\lambda_1$
and
$m_{\lambda_1}(\lambda)$
, we divide A(n) to
\begin{align*}A(n) = A_0(n)+ A_1(n) +A_2(n)+ A_3(n),\end{align*}
where
\begin{align*} A_0(n)\,:\!= \sum_{\substack{|\lambda|=n \\ \lambda_1 \leqslant y_0}} a(\lambda), \qquad\qquad A_1(n) \,:\!=\sum_{\substack{|\lambda|=n \\ \lambda_1 \gt y_0 \\ m_{\lambda_1}(\lambda)=1}} a(\lambda),\end{align*}
\begin{align*}A_2(n)\,:\!= \sum_{\substack{|\lambda|=n \\ \lambda_1 \gt y_0\\ m_{\lambda_1}(\lambda)= 2}} a(\lambda)\qquad \text{and}\qquad A_3(n)\,:\!= \sum_{\substack{|\lambda|=n \\ \lambda_1 \gt y_0\\ m_{\lambda_1}(\lambda)\geqslant 3}} a(\lambda).\end{align*}
Now we set, for each
$r\in \{0,1,2,3\}$
,
\begin{align*}\mathcal{B}_{\ell}^{(r)}\,:\!= \bigg\{ \sup_{X_{\ell -1}\lt n\leqslant X_{\ell}}\frac{|A_r(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}.\end{align*}
Note that
\begin{align*}\mathcal{B}_{\ell} \subset \bigcup_{r=0}^3\mathcal{B}_{\ell}^{(r)},\end{align*}
thus, to prove
$ \sum_{\ell \geqslant 1} \mathbb{P}\big[\mathcal{B}_{\ell}\big]$
converges, it suffices to prove
$ \sum_{\ell \geqslant 1} \mathbb{P}\big[\mathcal{B}_{\ell}^{(r)}\big]$
converges, for all
$r\in\{0,1,2,3\}$
.
Remark 3·1. The primary expression is A1, while all other terms Ar for r≠1 can be bounded with a more than sufficient estimate.
4. Convergence of
$\sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(0)}] \text{ and }$
$\sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(3)}]$
.
We start by looking at
$\mathcal{B}_{\ell}^{(0)}$
.
Lemma 4·1. The sum
$ \sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(0)}]$
converges.
Proof. We have
\begin{equation} \begin{aligned}\mathbb{E}\big[ |A_0(n)|^2\big] = \sum_{\substack{ |\lambda|=n \\ \lambda_1 \leqslant y_0}} \mathbb{E}\big[|a(\lambda)|^2\big] = \sum_{\substack{ |\lambda|=n \\ \lambda_1 \leqslant y_0}}\prod_k \frac{1}{k^{m_k}m_k!}.\end{aligned}\end{equation}
Arguing as Soundararajan and Zaman in [Reference Soundararajan and Zaman7], the right-hand side of (4·1) is the coefficient of
$z^n$
in the generating function
$\exp\big(\sum_{k\leqslant y_0}z^k/k \big)$
. Since the coefficients of this generating function are all non-negative, for any
$r \gt 0$
we conclude that
\begin{equation} \mathbb{E}\big[ |A_0(n)|^2\big] = \sum_{\substack{ |\lambda|=n \\ \lambda_0 \leqslant y_0}} \prod_k \frac{1}{k^{m_k}m_k!} \leqslant \frac{1}{r^n}\exp\bigg(\sum_{k\leqslant y_0}\frac{r^{k}}{k} \bigg).\end{equation}
By choosing
$r=\exp(1/y_0)$
and since
$X_{\ell-1}\lt n\leqslant X_{\ell}$
, we get
\begin{align*} \mathbb{E}\big[ |A_0(n)|^2\big] & \leqslant \frac{\exp\bigg(\sum_{k \leqslant y_0} \frac{{\rm e}^{k/y_0}}{k}\bigg)}{{\rm e}^{n/y_0}} \\ & \leqslant \frac{\exp\bigg(\sum_{k \leqslant y_0} \frac{{\rm e}}{k}\bigg)}{\exp(X_{\ell-1}/y_0)} \\ & \leqslant \frac{\exp\big(2{\rm e} \log y_0\big)}{\exp(X_{\ell-1}/y_0)} \\ & \leqslant \frac{y_0^{2{\rm e}}}{\exp(X_{\ell-1}/y_0)} \leqslant \frac{2^{2{\rm e}\ell^K}}{\exp\big(2^{c\, \ell^{K-2}}\big)},\end{align*}
where c is an absolute constant. Thus, by Markov’s inequality
\begin{align*}\mathbb{P}[\mathcal{B}_{\ell}^{(0)}]\leqslant \sum_{X_{\ell-1}\lt n\leqslant X_{\ell}} \frac{1}{(\log n)^{1/2+2\varepsilon}} \mathbb{E}\big[ |A_0(n)|^2\big] \leqslant \frac{2^{(2{\rm e}+1)\ell^K}}{\exp\big(2^{c\,\ell^{K-2}}\big)}.\end{align*}
It follows that the sum
$ \sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(0)}]$
converges.
Lemma 4·2. The sum
$ \sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(3)}]$
converges.
Proof. We have
\begin{align*} \mathbb{E}\big[ |A_3(n)|^2\big] & = \sum_{\substack{|\lambda|=n \\ m_{\lambda_1}(\lambda) \geqslant 3 \\ \lambda_1 \gt y_0} } \mathbb{E}\big[ |a(\lambda)|^2\big] \\ & \leqslant \sum_{ y_0 \lt k\leqslant n/3} \frac{1}{k^3} \sum_{\substack{|\lambda|=n-3k \\ \lambda_1 \leqslant k} }\mathbb{E}\big[ |a(\lambda)|^2\big].\end{align*}
Since, for every
$k,n\geqslant 1$
\begin{align*}\sum_{\substack{|\lambda|=n-3k \\ \lambda_1 \leqslant k} }\mathbb{E}\big[ |a(\lambda)|^2\big] \leqslant \sum_{\substack{|\lambda|=n-3k } }\mathbb{E}\big[ |a(\lambda)|^2\big] \leqslant 1, \end{align*}
we get
\begin{align*}\mathbb{E}\big[ |A_3(n)|^2\big] \leqslant \sum_{ y_0 \lt k\leqslant n/3} \frac{1}{k^3} \ll \frac{1}{y_0^2}.\end{align*}
Thus,
\begin{align*}\mathbb{P}[\mathcal{B}_{\ell}^{(3)}] \leqslant \sum_{X_{\ell-1} \lt n \leqslant X_{\ell}} \frac{1}{(\log n)^{1/2+2\varepsilon}}\mathbb{E}\big[ |A_3(n)|^2\big] \ll 2^{\ell^K} \frac{2^{2K\ell^{K-1}}}{2^{2\ell^K}} = \frac{2^{2K\ell^{K-1}}}{2^{\ell^K}}.\end{align*}
It follows that the sum
$ \sum_{\ell \geqslant 1} \mathbb{P}[\mathcal{B}_{\ell}^{(3)}]$
converges.
5. Upper bound of
$\mathbb{P}[\mathcal{B}^{(1)}_{\ell}]$
.
In this subsection, we give a bound of
$ \mathbb{P}[\mathcal{B}_{\ell}^{(1)}]$
. We consider the filtration
$\big\{ \mathcal{F}_k \big\}_{k\geqslant 1}$
, where
$\mathcal{F}_k$
is the
$\sigma$
-algebra generated by
$\{X(1),X(2),...,X(k-1)\}$
. We set the convention
$a(\lambda)=0$
for every
$|\lambda|\lt0$
. We have
\begin{align*}A_1(n)= \sum_{\substack{|\lambda|=n \\ \lambda_1 \gt y_0 \\ m_{\lambda_1}(\lambda)=1}} a(\lambda) = \sum_{ y_0 \lt k \leqslant n} \frac{X(k)}{\sqrt{k}} \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda).\end{align*}
Note that
$\sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda)$
depends only on X(i) for
$i\lt k$
. Note, as well
\begin{align*}\mathbb{E}\bigg[ \frac{X(k)}{\sqrt{k}} \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda)\, \bigg| \mathcal{F}_{k} \,\bigg]=\mathbb{E}\bigg[ \frac{X(k)}{\sqrt{k}} \, \bigg| \mathcal{F}_{k} \,\bigg] \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) =0.\end{align*}
Thus
$A_1(n)$
, as defined in Section 3, is a sum of martingale differences. Recall that
$y_j\asymp y_{j-1}$
, we set
\begin{equation} V(n) \,:\!= \sum_{ y_0 \lt k \leqslant n} \frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2.\end{equation}
We define
\begin{equation} \widetilde{V}(n)\,:\!= \sum_{\substack{ 1 \leqslant j\leqslant J \\ \frac{n}{y_{j}} \gt\ell^{100K}}}\sum_{ y_{j-1} \lt k \leqslant y_j} \frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2\end{equation}
and we set
\begin{equation} V(n,y_{j})\,:\!= \frac{1}{y_{j}}\sum_{ y_{j-1} \lt k \leqslant y_j} \bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2.\end{equation}
Note that the number of j such that
$n\geqslant y_j$
and
${n}/{y_{j}} \leqslant \ell^{100K}$
is less than
$100K \ell \log \ell+1$
. We have then
\begin{align*}\begin{aligned} V(n) & \leqslant \widetilde{V}(n) + (100K \ell \log \ell+1) \sup_{\substack{ \frac{n}{y_{j}} \leqslant \ell^{100K} \\ n \geqslant y_j}} V(n,y_{j}) \\ & \leqslant C_0\bigg( \widetilde{V}(n)+ \ell \log \ell \sup_{\substack{ 1\leqslant j\leqslant J}} V(n,y_{j}) \bigg),\end{aligned}\end{align*}
where
$C_0$
is a constant depend only on K. Let
\begin{equation} T(\ell) =\ell^{10} \,\,\,\,\,\text{ and }\,\,\,\,\, T_1(\ell)\,:\!= \frac{T(\ell)}{\ell \log\ell}.\end{equation}
We set the events
\begin{equation} \mathcal{T}(\ell)=:\mathcal{T} = \bigg\{ \sup_{ X_{\ell-1}\lt n \leqslant X_{\ell}} V(n) \leqslant 2C_0T(\ell)\ell^{K/2} \bigg\}\end{equation}
and
\begin{equation} \mathcal{T}_n=\mathcal{T}_n(\ell) \,:\!= \bigg\{ V(n) \leqslant 2C_0T(\ell)\ell^{K/2} \bigg\}.\end{equation}
We define finally the following probabilities
\begin{equation} \mathbb{P}^{(1)}_{\ell} \,:\!= \mathbb{P}\bigg[ \sup_{\substack{ X_{\ell-1} \lt n\leqslant X_{\ell} \\1 \leqslant j\leqslant J }} V(n,y_j)\gt T_1(\ell)\ell^{K/2} \bigg]\end{equation}
and
\begin{equation} \widetilde{\mathbb{P}}^{(1)}_{\ell} \,:\!= \mathbb{P}\bigg[ \sup_{\substack{ X_{\ell-1} \lt n\leqslant X_{\ell} }} \widetilde{V}(n) \gt T(\ell)\ell^{K/2} \bigg].\end{equation}
It is clear that
$ \mathbb{P}[\overline{\mathcal{T}}] \leqslant \mathbb{P}^{(1)}_{\ell} +\widetilde{\mathbb{P}}^{(1)}_{\ell}$
, where
$ \overline{\mathcal{T}}$
is the complement of
$ \mathcal{T}$
in sample space.
5·1. Bounding
$\widetilde{\mathbb{P}}^{(1)}_{\ell} $
.
The objective of this section is to establish the convergence of the summation
$\sum_{\ell \geqslant 1} \widetilde{\mathbb{P}}^{(1)}_{\ell}$
.
Lemma 5·1. The sum
$\sum_{\ell \geqslant 1} \widetilde{\mathbb{P}}^{(1)}_{\ell} $
converges.
Proof. Using the same argument as in the inequality (4·2), we have, for any
$r\gt 0$
\begin{align*} \mathbb{E}\bigg[ \bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2 \bigg] \leqslant \frac{1}{r^{n-k}} \exp\bigg(\sum_{m \lt k} \frac{r^m}{m} \bigg). \end{align*}
In particular for
$r={\rm e}^{1/k}$
, we have
\begin{align*} \mathbb{E}\bigg[ \bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2 \bigg] \leqslant \frac{k^6}{\exp\big(\frac{n-k}{k}\big)} =\frac{{\rm e}k^6}{\exp\big(\frac{n}{k}\big)} . \end{align*}
By using Markov’s inequality and the observation that
$T(\ell) \geqslant 1$
, we can derive a bound by utilising the inequality
$y_j\leqslant X_{\ell}^2$
, we get
\begin{align*} \begin{aligned} \widetilde{\mathbb{P}}^{(1)}_{\ell} &\leqslant \frac{1}{\ell^{K/2}} \sum_{X_{\ell-1}\lt n\leqslant X_{\ell}}\sum_{\substack{ 1 \leqslant j\leqslant J \\ \frac{n}{y_{j}} \gt\ell^{100K}}}\sum_{ y_{j-1} \lt k \leqslant y_j} \frac{1}{k}\mathbb{E}\bigg[ \bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2 \bigg] \\ & \leqslant \frac{1}{\ell^{K/2}} \sum_{X_{\ell-1}\lt n\leqslant X_{\ell}}\sum_{\substack{ 1 \leqslant j\leqslant J \\ \frac{n}{y_{j}} \gt\ell^{100K}}}\sum_{ y_{j-1} \lt k \leqslant y_j} \frac{{\rm e}k^5}{\exp\big(\frac{n}{k}\big)} \\ & \leqslant \frac{1}{\ell^{K/2}} \sum_{X_{\ell-1}\lt n\leqslant X_{\ell}}\sum_{\substack{ 1 \leqslant j\leqslant J \\ \frac{n}{y_{j}} \gt\ell^{100K}}}\sum_{ y_{j-1} \lt k \leqslant y_j} \frac{{\rm e}2^{10\ell^{K}}}{{\rm e}^{\ell^{100K}}} \\ & \ll \frac{\ell^{K/2} 2^{13\ell^K}}{{\rm e}^{\ell^{100K}}}. \end{aligned} \end{align*}
We deduce that the sum
$\sum_{\ell \geqslant 1} \widetilde{\mathbb{P}}^{(1)}_{\ell} $
converges.
5·2. Bounding
$\mathbb{P}^{(1)}_{\ell}$
.
The aim of this subsection is to give a bound of
$\mathbb{P}^{(1)}_{\ell}$
. We have for all
$1\leqslant j\leqslant J$
\begin{align*}V(n,y_j) \leqslant U_j\,:\!=\frac{1}{{y}_j} \sum_{r=0}^{+\infty}\max_{y_{j-1}\lt\beta\leqslant y_j} \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta}} a(\lambda)\bigg|^2\end{align*}
and
\begin{align*}V(n,y_0) \leqslant U_j\,:\!=\frac{1}{{y}_j} \sum_{r=0}^{+\infty} \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant y_0}} a(\lambda)\bigg|^2.\end{align*}
Let
\begin{align*}I_j\,:\!= \frac{1}{2\pi\widetilde{y}_j} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{0}}\bigg)^{-1/\ell^K}\int_{0}^{2\pi } \big|F_{y_j}({\rm e}^{i\vartheta})\big|^2{\rm d}\vartheta\end{align*}
where
$\widetilde{y}_j$
is defined in (3·1). Define
$\mathcal{S}$
to be the event
$\big\{ I_{j} \leqslant {T_1(\ell)^{1/2}}/{\ell ^{K/2}} \text{ for all } 0 \leqslant j\leqslant J \big\}$
and
$\mathcal{S}_j\,:\!=\big\{ I_{j} \leqslant {T_1(\ell)^{1/2}}/{\ell ^{K/2}} \big\} $
. Now we have
\begin{align*}\begin{aligned} \mathbb{P}_{\ell}^{(1)} & \leqslant \mathbb{P}\bigg[ \bigcup_{0\leqslant j\leqslant J}\bigg\{U_j \geqslant \frac{T(\ell) \ell^{K/2} }{ \ell \log \ell} \bigg\} \bigcap \big\{ \mathcal{S}_{j-1}\big\}\bigg] + \mathbb{P}\big[\,\overline{\mathcal{S}}\,\big]\\ & \leqslant \sum_{j=0}^{J} \mathbb{P}\bigg[ \bigg\{U_j \geqslant \frac{T(\ell) \ell^{K/2} }{ \ell \log \ell} \bigg\} \bigcap \big\{ \mathcal{S}_{j-1}\big\}\bigg] + \mathbb{P}\big[\,\overline{\mathcal{S}}\,\big].\end{aligned}\end{align*}
Let start by treating
\begin{align*}\widetilde{\mathbb{P}}_j\,:\!=\mathbb{P}\bigg[ \bigg\{U_j \geqslant \frac{T(\ell) \ell^{K/2} }{ \ell \log \ell} \bigg\} \bigcap \big\{ \mathcal{S}_{j-1}\big\}\bigg].\end{align*}
By Markov’s inequality, we have
\begin{align*}\begin{aligned}\widetilde{\mathbb{P}}_j & \leqslant \mathbb{P}\bigg[ \bigg\{U_j \geqslant \frac{T(\ell) \ell^{K/2} }{ \ell \log \ell} \bigg\}\, \bigg| \, \big\{ \mathcal{S}_{j-1}\big\}\bigg]\\ & \leqslant \frac{\ell \log \ell}{T(\ell) \ell^{K/2}} \mathbb{E}\big[ U_j \, \big|\,\mathcal{S}_{j-1}\big].\end{aligned}\end{align*}
We will need the following lemmas.
Lemma 5·2. Let
$r\geqslant 1$
be a fixed integer. The sequence
$\bigg( \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta}} a(\lambda)\bigg|\bigg)_{\beta}$
is a submartingale with respect to the filtration
$\big(\mathcal{F}_{\beta}\big)_{\beta}$
, where
$\mathcal{F}_{\beta} $
is the
$\sigma$
-algebra generated by
$\big\{ X(k); k\leqslant \beta \big\}$
.
Proof. Let
$r\geqslant 1$
and
$\beta \geqslant 0$
. By using the fact that
$\int z^n {\rm d}\mathbb{P}=0$
for all
$n\geqslant 1$
(which is equivalent to say
$\mathbb{E}[(X(\beta+1))^n]=0$
), we have
\begin{align*} \begin{aligned} \mathbb{E}\bigg[\bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta+1}} a(\lambda)\bigg| \, \big| \, \mathcal{F}_{\beta}\bigg] & = \int \bigg|\sum_{n\geqslant 0} \bigg(\frac{z}{\sqrt{\beta+1}}\bigg)^n \sum_{\substack{|\lambda|=r-n(\beta+1) \\ \lambda_1 \leqslant \beta}} a(\lambda)\bigg| {\rm d}\mathbb{P} \\ & \geqslant \bigg| \int \sum_{n\geqslant 0} \bigg(\frac{z}{\sqrt{\beta+1}}\bigg)^n \sum_{\substack{|\lambda|=r-n(\beta+1) \\ \lambda_1 \leqslant \beta}} a(\lambda) {\rm d}\mathbb{P}\bigg| \\ & = \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta}} a(\lambda)\bigg|. \end{aligned} \end{align*}
This completes the proof.
Lemma 5·3. For
$\ell$
large enough, the sequence
$(I_j)_{j\geqslant 0}$
is supermartingale with respect to the filtration
$(\mathcal{F}_{y_j})_{j\geqslant 0}$
.
Proof. We have
\begin{align*} \begin{aligned} \mathbb{E}\big[I_{j} \, \big| \mathcal{F}_{y_{j-1}}\,\big] & = \frac{1}{2\pi\widetilde{y}_j} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{0}}\bigg)^{-1/\ell^K}\int_{0}^{2\pi } \mathbb{E}\bigg[ \big|F_{y_j}({\rm e}^{i\vartheta})\big|^2\, \big| \, \mathcal{F}_{y_j}\bigg]{\rm d}\vartheta \\ & = \frac{1}{2\pi\widetilde{y}_j} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{0}}\bigg)^{-1/\ell^K}\int_{0}^{2\pi } \mathbb{E}\bigg[ \bigg| \exp\bigg( \sum_{y_{j-1}\lt k\leqslant y_j}\frac{X(k)}{\sqrt{k}} {\rm e}^{ik\vartheta}\bigg)\bigg|^2\bigg]\big|F_{y_{j-1}}({\rm e}^{i\vartheta})\big|^2{\rm d}\vartheta \\ & = \frac{1}{2\pi\widetilde{y}_j} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{0}}\bigg)^{-1/\ell^K} \exp\bigg( \sum_{y_{j-1}\lt k\leqslant y_j}\frac{1}{k} \bigg)\int_{0}^{2\pi } \big|F_{y_{j-1}}({\rm e}^{i\vartheta})\big|^2{\rm d}\vartheta. \end{aligned} \end{align*}
In the sake of readability, we set
\begin{align*} b_j\,:\!= {\rm e}^{-1/\ell} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{j-1}}\bigg)^{-1/\ell^K} \exp\bigg( \sum_{y_{j-1}\lt k\leqslant y_j}\frac{1}{k} \bigg). \end{align*}
By collecting the previous computations together, we find
\begin{align*} \begin{aligned} \mathbb{E}\big[I_{j} \, \big| \mathcal{F}_{y_{j-1}}\,\big] \leqslant b_jI_{j-1}. \end{aligned} \end{align*}
To complete the proof it suffices to prove that
$b_j\leqslant 1$
. Recall that
${\widetilde{y}_j}/{\widetilde{y}_{j-1}}={\rm e}^{1/\ell}$
. For
$\ell$
large enough, we have
\begin{align*}\begin{aligned}b_j & = {\rm e}^{-1/\ell} \bigg(\frac{\widetilde{y}_{j}}{\widetilde{y}_{j-1}}\bigg)^{-1/\ell^K} \exp\bigg( \sum_{y_{j-1} \lt k\leqslant y_j}\frac{1}{k} \bigg)\\ & = \exp \bigg( -\frac{1}{\ell} + \frac{-1}{\ell^{K+1}}+\sum_{y_{j-1} \lt k \leqslant y_{j}} \frac{1}{k} \bigg) \\ & = \exp \bigg(\frac{-1}{\ell^{K+1}} +O\bigg(\frac{1}{y_0}\bigg) \bigg).\end{aligned}\end{align*}
Recall that
$y_0={2^{\ell^K}}/{2^{K\ell^{K-1}}}$
. Thus, for large
$\ell$
, we have
$b_j\leqslant 1$
. It follows then that
\begin{align*}\mathbb{E}\big[I_{j} \, \big| \,\mathcal{F}_{y_{j-1}}\big] \leqslant ~ I_{j-1}.\end{align*}
Using Lemma 5·2
\begin{align*}\begin{aligned} \mathbb{E}\big[ U_j \, \big|\,\mathcal{S}_{j-1}\big] & = \frac{1}{ y_j}\sum_{r=0}^{+\infty}\mathbb{E}\bigg[ \max_{y_{j-1}\lt\beta \leqslant y_j} \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta}} a(\lambda)\bigg|^2 \, \big|\,\mathcal{S}_{j-1}\bigg] \\& \leqslant \frac{4}{ y_j}\sum_{r=0}^{+\infty}\mathbb{E}\bigg[ \bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant y_j}} a(\lambda)\bigg|^2 \, \big|\,\mathcal{S}_{j-1}\bigg] \ll \mathbb{E}\big[I_j\,\big|\, \mathcal{S}_{j-1}\big].\end{aligned}\end{align*}
Now by using Lemma 5·3 we have
$\mathbb{E}\big[I_j\,\big|\, \mathcal{F}_{j-1}\big]\leqslant I_{j-1}.$
Thus we have
\begin{align*}\mathbb{E}\big[ U_j \, \big|\,\mathcal{S}_{j-1}\big] \ll \mathbb{E}\big[I_j\,\big|\, \mathcal{S}_{j-1}\big] = \mathbb{E}\bigg[\mathbb{E}\big[I_j\,\big|\, \mathcal{F}_{j-1}\big]\,\big|\, \mathcal{S}_{j-1}\bigg] \leqslant \mathbb{E}\big[I_{j-1}\,\big|\, \mathcal{S}_{j-1}\big]\leqslant \frac{T_1(\ell)^{1/2}}{\ell^{K/2}}.\end{align*}
Thus, we get at the end
\begin{equation} \sum_{j=1}^{J}{ \tilde{\mathbb{P}}}_j\ll \sum_{j=1}^{J} \frac{T_1(\ell)^{1/2}}{\ell^{K/2}} \frac{\ell \log \ell}{T(\ell) \ell^{K/2}} \ll \frac{T_1(\ell)^{1/2}\ell \log \ell}{T(\ell)}=\frac{\sqrt{\ell \log \ell}}{T(\ell)^{1/2}}.\end{equation}
Recall that
$T(\ell) = \ell^{10}$
(see (5·4)), we have
$\sum_{\ell \geqslant 1} {\sqrt{\ell \log \ell}}/{T(\ell)^{1/2}}$
converges.
We now look at
$ \mathbb{P}\big[ \mathcal{S}\big]$
.
We define
$\mathcal{A}\,:\!= \big\{ I_0\leqslant {T_1^{1/4}(\ell)}/{\ell^{K/2}} \big\}$
.
Lemma 5·4. For
$\ell$
large enough, we have
$\mathbb{P}\big[\,\overline{\mathcal{S}}\,\big]\ll {1}/{(T_1(\ell))^{1/6}}$
.
Proof. Indeed, we have
\begin{align*}\mathbb{P}\big[\,\overline{\mathcal{S}}\,\big] \leqslant \mathbb{P}\bigg[\max_{0 \leqslant j\leqslant J} I_j \gt \frac{(T_1(\ell))^{1/2}}{\ell^{K/2}}\, \bigg|\, \mathcal{A} \bigg] + \mathbb{P}\big[\,\overline{\mathcal{A}}\,\big].\end{align*}
Recall, by Lemma 5·3, the sequence
$(I_j)$
is supermartingale. Thus we have
\begin{align*}\mathbb{P}\bigg[\max_{0 \leqslant j\leqslant J} I_j \gt \frac{T_1^{1/2}(\ell)}{\ell^{K/2}}\, \bigg|\, \mathcal{A} \bigg] \leqslant \frac{\ell^{K/2}}{(T_{1}(\ell))^{1/2}} \mathbb{E}\big[I_0\, \big|\, \mathcal{A}\big] \leqslant \frac{1}{(T_{1}(\ell))^{1/4}}.\end{align*}
The second part of the lemma follows easily from Markov’s inequality and Lemma 2·3 for
$R=y_0$
and
$q=2/3$
. We then have
\begin{align*}\mathbb{P}\bigg[I_{0} \gt \frac{T_1(\ell)^{1/4}}{\ell ^{K/2}} \bigg]\leqslant \frac{\ell ^{\frac{K}{3}}\mathbb{E}[I_{0}^{\frac{2}{3}}]}{T_1(\ell)^{\frac{1}{6}}} \ll \frac{1}{T_1(\ell)^{1/6}}. \end{align*}
Proposition 5·5.
$ \sum_{\ell \geqslant 1 }\mathbb{P}^{(1)}_{\ell} $
converges.
5·3. Convergence of
$\sum_{\ell \geqslant 1} \mathbb{P}\big[ \mathcal{B}^{(1)}_\ell\big]$
.
In this section, we follow the same steps as in [Reference Caich1, section 6·8].
Proposition 5·6. The sum
$\sum_{\ell \geqslant 1} \mathbb{P}\big[ \mathcal{B}^{(1)}_\ell\big]$
converges.
Proof. We have
\begin{align*} \begin{aligned} \mathbb{P}\big[ \mathcal{B}^{(1)}_\ell\big] & \leqslant \mathbb{P}\bigg[\bigcup_{X_{\ell-1} \lt n \leqslant X_{\ell}} \bigg\{ \frac{|A_1(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n \bigg]+ \mathbb{P}\big[\,\overline{\mathcal{T}} \,\big] \\ & \leqslant \sum_{X_{\ell-1} \lt n \leqslant X_{\ell}} \mathbb{P}\bigg[ \bigg\{ \frac{|A_1(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n \bigg] + \mathbb{P}\big[\,\overline{\mathcal{T}} \,\big]. \end{aligned} \end{align*}
By Proposition 5·5 and Lemma 5·1, the sum
$\sum_{\ell \geqslant 1}\mathbb{P}\big[\,\overline{\mathcal{T}(\ell)} \,\big]$
converges. By applying Lemma 2·2, and since by assumption
$K\varepsilon = 25$
, we have
\begin{align*} \begin{aligned} \mathbb{P}\bigg[ \bigg\{ \frac{|A_r(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n \bigg] & \leqslant 2 \exp \bigg(\frac{-C_2 \ell^{3K/2+2\varepsilon K}}{\ell^{K/2}T(\ell)} \bigg) \\ &\leqslant 2 \exp \bigg(-C_2 \ell^{K+44}\bigg), \end{aligned} \end{align*}
where
$C_2\gt 0$
is an absolute constant. Finally we get
\begin{align*} \mathbb{P}\big[\mathcal{B}_{\ell}^{(1)}\big] \ll \exp \bigg( \log 2\ell^{K}-C_2\, \ell^{K} \ell^{44} \bigg) + \mathbb{P}\big[\,\overline{\mathcal{T}} \,\big]. \end{align*}
Thus the sum
$\sum_{\ell \geqslant 1}\mathbb{P}\big[\mathcal{B}_{\ell}^{(1)}\big]$
converges.
6. Upper bound of
$\mathbb{P}[\mathcal{B}^{(2)}_{\ell}]$
.
In this subsection, we give an upper bound of
$\mathbb{P}[\mathcal{B}_{\ell}^{(2)}]$
.
6·1. Preliminaries.
We start with some results.
\begin{align*}A_2(n)= \sum_{\substack{|\lambda|=n \\ \lambda_1 \gt y_0 \\ m_{\lambda_1}(\lambda)=2}} a(\lambda) = \sum_{ y_0 \lt k \leqslant n/2} \frac{X(k)^2}{k} \sum_{\substack{|\lambda|=n-2k \\ \lambda_1 \lt k }} a(\lambda).\end{align*}
Note, as in Section 5,
$A_2(n)$
is a sum of martingale difference with respect to the same filtration
$(\mathcal{F}_{k})_{k\geqslant 1}$
. By following the same steps as in Section 5, the problem is reduced to studying:
\begin{align*} W(n) &\,:\!= \sum_{ y_0 \lt k \leqslant n/2} \frac{1}{2k^2}\bigg| \sum_{\substack{|\lambda|=n-2k \\ \lambda_1 \lt k }} a(\lambda) \bigg|^2 \\ & \leqslant \frac{1}{2y_0} \sum_{ y_0 \lt k \leqslant n} \frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k/2 }} a(\lambda) \bigg|^2.\end{align*}
We set
\begin{align*}V^{(2)}(n)\,:\!= \sum_{ y_0 \lt k \leqslant n} \frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k/2 }} a(\lambda) \bigg|^2.\end{align*}
One can see that
$V^{(2)}(n)$
is similar to V(n), introduced in (5·1), with a little difference over the sum. We define the analogues of
$\widetilde{V}(n)$
and
$V(n,y_j)$
:
\begin{align*}\widetilde{V}^{(2)}(n)\,:\!= \sum_{\substack{ 1 \leqslant j\leqslant J \\ \frac{n}{y_{j}} \gt\ell^{100K}}}\sum_{ y_{j-1} \lt k \leqslant y_j} \frac{1}{k}\bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k/2 }} a(\lambda) \bigg|^2\end{align*}
and
\begin{align*}V^{(2)}(n,y_{j})\,:\!= \frac{1}{y_{j}}\sum_{ y_{j-1} \lt k \leqslant y_j} \bigg| \sum_{\substack{|\lambda|=n-k \\ \lambda_1 \lt k/2 }} a(\lambda) \bigg|^2.\end{align*}
We then have, as in (5)
\begin{align*}\begin{aligned} V^{(2)}(n) \leqslant C_0\bigg( \widetilde{V}^{(2)}(n)+ \ell \log \ell \sup_{\substack{ 1\leqslant j\leqslant J}} V^{(2)}(n,y_{j}) \bigg),\end{aligned}\end{align*}
where
$C_0$
is an absolute constant. We set the events
\begin{equation} \mathcal{T}^{(2)}= \mathcal{T}^{(2)}(\ell)\,:\!= \bigg\{ \sup_{ X_{\ell-1}\lt n \leqslant X_{\ell}} V^{(2)}(n) \leqslant 2C_0T(\ell)\ell^{K/2} \bigg\}\end{equation}
and
\begin{equation} \mathcal{T}^{(2)}_n= \mathcal{T}^{(2)}_n(\ell) \,:\!= \bigg\{ V^{(2)}(n) \leqslant 2C_0T(\ell)\ell^{K/2} \bigg\}.\end{equation}
We define finally the analogue probabilities as in (5·7) and (5·8)
\begin{equation} \mathbb{P}^{(2)}_{\ell} \,:\!= \mathbb{P}\bigg[ \sup_{\substack{ X_{\ell-1} \lt n\leqslant X_{\ell} \\1 \leqslant j\leqslant J }} V^{(2)}(n,y_j)\gt T_1(\ell)\ell^{K/2} \bigg]\end{equation}
and
\begin{equation} \widetilde{\mathbb{P}}^{(2)}_{\ell} \,:\!= \mathbb{P}\bigg[ \sup_{\substack{ X_{\ell-1} \lt n\leqslant X_{\ell} }} \widetilde{V}^{(2)}(n) \gt T(\ell)\ell^{K/2} \bigg].\end{equation}
It is clear that
$ \mathbb{P}[\overline{\mathcal{T}}] \leqslant \mathbb{P}^{(2)}_{\ell} +\widetilde{\mathbb{P}}^{(2)}_{\ell}$
.
Lemma 6·1. The sum
$\sum_{\ell \geqslant 1} \widetilde{\mathbb{P}}^{(2)}_{\ell} $
converges.
Proof. This is the same as the proof of Lemma 5·1.
On the other hand, we have
\begin{align*}V^{(2)}(n,y_j) \ll U^{(2)}_j\,:\!=\frac{1}{y_j} \sum_{r=0}^{+\infty} \max_{y_{j-1}\lt\beta \leqslant y_j}\bigg| \sum_{\substack{|\lambda|=r \\ \lambda_1 \leqslant \beta/2}} a(\lambda)\bigg|^2.\end{align*}
By following exactly the same steps as in Section 5·2 we get the analogue of Proposition 5·5.
Lemma 6·2. For sufficiently large
$\ell$
, we have
\begin{equation} \mathbb{P}_{\ell}^{(2)} \ll \frac{1}{T_1(\ell)^{1/6}}.\end{equation}
Furthermore, since
$T_1(\ell)\gg \ell^8$
, the sum
$\sum_{\ell \geqslant 1} \mathbb{P}_{\ell}^{(2)} $
converges.
In addition, we have:
Lemma 6·3. The sum
$\sum_{\ell \geqslant 1}\mathbb{P}\big[\,\overline{\mathcal{T}^{(2)}(\ell)} \,\big]$
converges.
6·2. Convergence of
$\sum_{\ell \geqslant 1} \mathbb{P}\big[ \mathcal{B}^{(2)}_\ell\big]$
.
This section is similar to section 5·3.
Proposition 6·4.
The sum
$\sum_{\ell \geqslant 1} \mathbb{P}\big[ \mathcal{B}^{(2)}_\ell\big]$
converges.
Proof. We have
\begin{align*} \begin{aligned} \mathbb{P}\big[ \mathcal{B}^{(2)}_\ell\big] & \leqslant \mathbb{P}\bigg[\bigcup_{X_{\ell-1} \lt n \leqslant X_{\ell}} \bigg\{ \frac{|A_2(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n^{(2)} \bigg]+ \mathbb{P}\big[\,\overline{\mathcal{T}^{(2)}} \,\big] \\ & \leqslant \sum_{X_{\ell-1} \lt n \leqslant X_{\ell}} \mathbb{P}\bigg[ \bigg\{ \frac{|A_2(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n^{(2)} \bigg] + \mathbb{P}\big[\,\overline{\mathcal{T}^{(2)}} \,\big]. \end{aligned} \end{align*}
Recall that
$T(\ell)=\ell^{10}$
, which gives the convergence of the sum
$\sum_{\ell \geqslant 1}\mathbb{P}\big[\,\overline{\mathcal{T}^{(2)}(\ell)} \,\big]$
by Lemma 6·3. By applying Lemma 2·2, and since
$W(n)\leqslant V^{(2)}(n)/y_0$
, we have
\begin{align*} \begin{aligned} \mathbb{P}\bigg[ \bigg\{ \frac{|A_2(n)|}{(\log n )^{3/4 + \varepsilon}}\gt 1 \bigg\}\bigcap \mathcal{T}_n^{(2)} \bigg] & \leqslant 2 \exp \bigg(\frac{-C_2 y_0\ell^{3K/2+2\varepsilon K}}{\ell^{K/2}T(\ell)} \bigg) \\ &\leqslant 2 \exp \bigg(-C_2 y_0 \ell^{K+44}\bigg), \end{aligned} \end{align*}
where
$C_2\gt 0$
is an absolute constant. We have at the end
\begin{align*} \mathbb{P}\big[\mathcal{B}_{\ell}^{(2)}\big] \ll \exp \bigg( \log 2\ell^{K}-C_2\,y_0 \ell^{K} \ell^{44} \bigg) + \mathbb{P}\big[\,\overline{\mathcal{T}^{(2)}} \,\big]. \end{align*}
Thus the sum
$\sum_{\ell \geqslant 1}\mathbb{P}\big[\mathcal{B}_{\ell}^{(2)}\big]$
converges.
Acknowledgements
The author would like to thank his supervisor Régis de la Bretèche for his patient guidance, encouragement and the judicious advice he provided throughout the work that led to this paper.
