# Implications of subconvexity bounds for the moments of $\zeta(s)$

Kevin Smith

###### Abstract.

It is well-known that upper bounds for moments of the Riemann zeta function
$\zeta(s)$ have implications for subconvexity bounds. In this paper we explore
some implications in the opposite direction using functional analysis in the
right-half of the critical strip. The main results characterise potential
transitions in the behaviour of the moments.

##  1\. Introduction

Since Montgomery’s work [16] on the pair-correlation of the zeros of the
Riemann zeta function $\zeta(s)$, in particular the work of Katz and Sarnak
[14], Keating and Snaith [15], Conrey, Farmer, Keating, Rubinstein and Snaith
[1] and Farmer, Gonek and Hughes [5], it is now widely believed that zero
distributions, moments and bounds in families of $L$-functions approximating
critical lines and critical points correspond to those of the characteristic
polynomials of large random matrices in one of the classical compact groups.
For the moments of $\zeta(s)$,

$\displaystyle
M_{k}(\sigma,T)=\int_{0}^{T}\left|\zeta\left(\sigma+it\right)\right|^{2k}dt,$

it was conjectured (assuming the Riemann hypothesis) by Keating and Snaith
[15] by analogy with the unitary case, and by Conrey and Ghosh [4] and Conrey
and Gonek [3] using conjectures on additive divisor sums, that

(1.1) $\displaystyle M_{k}(1/2,T)\sim a_{k}g_{k}T(\log T)^{k^{2}}$

where

(1.2) $\displaystyle a_{k}=\prod_{p\textrm{
prime}}\left(1-\frac{1}{p}\right)^{(k-1)^{2}}\sum_{n=0}^{\infty}{k-1\choose
n}^{2}p^{-n}\hskip 11.38092pt\textrm{and }\hskip
11.38092ptg_{k}=\frac{G^{2}(k+1)}{G(2k+1)}$

in which $G(z)$ is Barnes’ $G$-function. In particular, the work of Keating
and Snaith and Gonek, Hughes and Keating [7] predicts that the factor
$g_{k}(\log T)^{k^{2}}$ arises from the influence of the zeros and their
distribution on the critical line $\sigma=1/2$, while the constant factor
$a_{k}$ arises from the influence of the primes. More recently, building on
work of Soundararajan [17], Harper [10] has shown that the Riemann hypothesis
alone implies that111The lower bound here holds unconditionally.

(1.3) $\displaystyle M_{k}(1/2,T)\asymp_{k}T(\log T)^{k^{2}}\hskip
28.45274pt(k>0).$

The upper bound in (1.3) is known unconditionally only when $k\leq 2$. The
cases $k=1$ and $2$ are classical results of Hardy and Littlewood [8] and
Ingham [12], respectively. The difficulties involved in computing
$M_{k}(\sigma,T)$ increase not only with increasing $k$ on the critical line,
but also as $\sigma$ decreases toward $1/2$ for each fixed $k\in\mathbb{N}$.
However, in the region $1/2<\sigma<1$ where zeros are rarer (of course, the
Riemann hypothesis predicts there are none), it is expected that
$M_{k}(\sigma,T)$ is completely determined by the influence of the primes.
Indeed, this is a consequence of the Lindelöf hypothesis which we discuss
below.

Denoting by $d_{k}(n)$ the number of ways of writing $n$ as a product of $k$
factors, $M_{k}(\sigma,T)$ obeys a convexity principle as a function of
$\sigma$, one consequence of which is that the statements

(1.4) $\displaystyle M_{k}(\sigma_{k},T)\ll T^{1+\epsilon}\hskip
28.45274pt(\sigma_{k}\geq 1/2),$ (1.5) $\displaystyle M_{k}(\sigma,T)\ll
T\hskip 28.45274pt(1/2\leq\sigma_{k}<\sigma<1)$

and

(1.6)
$\displaystyle\lim_{T\rightarrow\infty}\frac{M_{k}(\sigma,T)}{T}=\sum_{n=1}^{\infty}\frac{d^{2}_{k}(n)}{n^{2\sigma}}\hskip
28.45274pt(1/2\leq\sigma_{k}<\sigma<1)$

are equivalent (Titchmarsh [18], Section 7.9). Another consequence is that for
every $k\in\mathbb{N}$ there is a $\sigma_{k}<1$ with the above property.
General results of this type are given by Ivíc ([13], Section 8.5). However,
beyond the theorems of Hardy and Littlewood and Ingham mentioned above which
yield $\sigma_{1}=\sigma_{2}=1/2$, currently the best known estimates are
$\sigma_{3}\leq 7/12$ (due to Ivíc cited above), $\sigma_{4}\leq 5/8$ and
$M_{3}(1/2,T)\ll_{\epsilon}T^{5/4+\epsilon}$. The latter two results are
consequences of Heath-Brown’s twelfth moment estimate [11].

Subconvexity bounds are described by the continuous non-negative non-
increasing convex function

$\displaystyle\mu(\sigma)=\inf\\{\mu:\zeta(\sigma+it)\ll t^{\mu}\\}.$

Here, the Lindelöf hypothesis asserts that $\mu(1/2)=0$ (currently the best
known bound is $\mu(1/2)<13/84$ due to Bourgain [2]). In terms of moments, the
Lindelöf hypothesis is equivalent to the statement that
$M_{k}(1/2,T)\ll_{k,\epsilon}T^{1+\epsilon}$ for all $k\in\mathbb{N}$, i.e.

(1.7) $\displaystyle\mu(1/2)=0\Leftrightarrow\sigma_{k}=1/2\hskip
14.22636pt(k\in\mathbb{N})$

due to Hardy and Littlewood [9] and, since (1.4), (1.5) and (1.6) are
equivalent, it is evident that the Lindelöf hypothesis depends only on the
moments in the region $\sigma>1/2$. In fact, it is a consequence of quite
general properties of $\zeta(s)$ that the implication (1.7) has a “pointwise”
analogue in one direction at least, i.e.

(1.8) $\displaystyle\sigma_{k}=1/2\Rightarrow\mu(1/2)<1/2k$

(for instance, see Ivíc ([13], Section 8.1) or Heath-Brown’s notes ([18],
Chapter 7). Yet, beyond that which can be derived as a consequence of
convexity, i.e. results of the type given by Ivíc ([13], Section 8.5), the
implications of

(1.9) $\displaystyle\mu(1/2)<1/2k$

for the numbers $\sigma_{k}$ are unclear. In particular, the converse
statement of (1.8) is not known. Similarly, despite that the inverse statement
$\sigma_{k}>1/2$ certainly implies that $\mu(1/2)>0$, in other words

(1.10)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{M_{k}(\sigma,T)}{T}=\infty\hskip
28.45274pt(1/2<\sigma\leq\sigma_{k}),$

it is not apparent what lower bound for $\mu(1/2)$ could be given if
$\sigma_{k}>1/2$ were true for some $k\in\mathbb{N}$.

In the absence of further information in this direction, in this paper we
assume (1.9) and establish some results of a probabilistic character. The main
idea in our work is that (1.9) permits us to perform a sort of “Fourier
analysis” that leads to particular insights regarding those values of
$1/2<\sigma<1$ for which $M_{k}(\sigma,T)\ll_{k}T$, or not. These are
contained in theorems 1 and 2 below.

We denote by

$\displaystyle|S|=\lim_{T\rightarrow\infty}\frac{\textrm{meas}\left(S\cap[0,T]\right)}{T}$

the natural density of Lebesgue measurable subsets $S\subseteq\mathbb{R}^{+}$
(if $|S|=0$ we say that $S$ is null) and $\textrm{Pr}\left(|g|\geq
x\right)=|X|$ where $X\subseteq\mathbb{R}^{+}$ is the subset on which $|g|\geq
x$. To begin, we observe that the trivial bound

$\displaystyle\frac{\left|\int_{0}^{T}f^{2}gdt\right|}{\int_{0}^{T}|f|^{2}dt}\leq\textrm{ess
sup}_{[0,T]}|g|$

may be significantly improved when $T$ is large if $f$ is not concentrated on
a null set, by which we mean the following.

######  Proposition 1 (Concentration on a null set).

The following statements are equivalent. If $f$ satisfies them, we say that
$f$ is not concentrated on a null set (not CNS).

  * (_a_)

If $g$ is bounded then

(1.11)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\left|\int_{0}^{T}f^{2}gdt\right|}{\int_{0}^{T}|f|^{2}dt}\leq\sup\\{x:\Pr(|g|\geq
x)>0\\}.$

  * (_b_)

If $S$ is null then

(1.12)
$\displaystyle\int_{S\cap[0,T]}|f|^{2}dt=o\left(\int_{0}^{T}|f|^{2}dt\right)\hskip
28.45274pt(T\rightarrow\infty).$

Moreover, if $\int_{0}^{T}|f|^{2}dt\gg T$, we may include

  * (_c_)

If $S$ is null then for every $\epsilon>0$ there is a bounded function
$f_{N,S}$ $(N=N(\epsilon))$ such that

(1.13)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\int_{S\cap[0,T]}|f-f_{N,S}|^{2}dt}{\int_{0}^{T}|f|^{2}dt}<\epsilon.$

If also $\int_{0}^{T}|f|^{2}dt\ll T$ then condition _(_ c _)_ may be replaced
with

  * (_d_)

For every $\epsilon>0$ there is a bounded function $f_{N}$ $(N=N(\epsilon))$
such that

(1.14)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}|f-f_{N}|^{2}dt<\epsilon.$

######  Theorem 1.

Assume $\mu(1/2)<1/2k$ and $1/2<\sigma<1$. Then $\zeta^{k}(\sigma+it)$ is not
CNS if and only if $\sigma_{k}<\sigma<1$.

Thus, Theorem 1 shows that if $\mu(1/2)<1/2k$ and $\sigma_{k}>1/2$ then the
density

(1.15) $\displaystyle\frac{|\zeta(\sigma+it)|^{2k}}{M_{k}(\sigma,T)}\hskip
28.45274pt(t\in[0,T],k\in\mathbb{N})$

must undergo a critical transition at the point $\sigma_{k}$ in the sense that
(1.12) holds with $f(t)=|\zeta(\sigma+it)|^{k}$ for $\sigma_{k}<\sigma<1$ and
fails for $1/2<\sigma\leq\sigma_{k}$. Since we know that $\mu(1/2)<1/6$ and
$1/2\leq\sigma_{3}\leq 7/12$, this suggests an alternative way to try to rule
out the possibility that $\sigma_{3}>1/2$. Theorem 1 arises as a special case
of the more general statement Proposition 5 given in Section 2, which captures
the influence of a broader class of locally integrable functions being
concentrated on a null set. It is here that functional analysis plays a key
role.

One explanation for the apparent disparity between bounds for $\mu(1/2)$ and
bounds for $\sigma_{k}$, and indeed our ability to compute the moments, is
that such atypical large values of $|\zeta(\sigma+it)|^{k}$ (if they exist)
must occur “near” the imaginary ordinates of zeros away from the critical line
(if they exist). This is clear from, for example, the proof of Proposition 2
below. Yet, the available estimates for the density of zeros decay (as a
function of $\sigma$) too slowly for us to draw the inverse or converse
conclusions from them alone. Here several different types of estimate for the
number $N(\sigma,T)$ of zeros $\rho=\beta+i\gamma$ with $\beta\geq\sigma$ and
$|\gamma|\leq T$ are available ([18], Section 9.15), but these typically
involve functions $\alpha(\sigma)$ and $\beta(\sigma)$ such that

(1.16) $\displaystyle N(\sigma,T)\ll T^{\alpha(\sigma)}(\log
T)^{\beta(\sigma)}$

in which $\alpha(\sigma)$ decays slowly in the range $\sigma>1/2$ with
$\alpha(1/2)=1$. Nonetheless, these zero density estimates are sufficient to
establish Proposition 2 below. Here we set $\theta_{P_{N}}=\arg P_{N}$ where

(1.17) $\displaystyle P_{N}(\sigma+it)=\exp\left(\sum_{2\leq n\leq
N^{2}}\frac{\Lambda_{N}(n)}{n^{\sigma+it}\log n}\right),$ (1.20)
$\displaystyle\Lambda_{N}(n)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\Lambda(n)\hskip 113.81102pt(n\leq
N)\\\ \Lambda(n)\left(2-\frac{\log n}{\log N}\right)\hskip
54.06006pt(\textrm{otherwise})\end{array}\right.$

in which $\Lambda(n)$ is the Von-Mangoldt function

(1.23) $\displaystyle\Lambda(n)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\log p\hskip
36.98866pt(n=p^{m},p\textrm{ prime},m\in\mathbb{N})\\\ 0\hskip
54.06006pt(\textrm{otherwise}).\end{array}\right.$

We also set $Z_{N}=\zeta/P_{N}$ and

$\displaystyle\theta_{Z_{N}}=\arg Z_{N}$

so $\theta_{Z_{N}}=\theta_{\zeta}-\theta_{P_{N}}$ $\pmod{(-\pi,\pi]}$ and

$\displaystyle|\zeta|^{2k}e^{2ik\theta_{Z_{N}}}=|P_{N}|^{2k}Z_{N}^{2k}=\zeta^{2k}e^{-2ik\theta_{P_{N}}}\hskip
28.45274pt(k\in\mathbb{N}).$

######  Proposition 2.

For every $1/2<\sigma<1$ and $\epsilon>0$

$\displaystyle\textrm{\emph{Pr}}\left(\left|\theta_{Z_{N}}(\sigma+it)\pmod{(-\pi,\pi]}\right|\geq\epsilon\right)=0\hskip
28.45274pt(N>N(\epsilon)).$

Proposition 2 plays a key role in our proofs because it implies that
$f=e^{ik\theta_{\zeta}}$ satisfies (1.14) with $f_{N}=e^{ik\theta_{P_{N}}}$
for $1/2<\sigma<1$ and $k\in\mathbb{N}$.

Our second theorem demonstrates a “zero-one” law for the density (1.15) in the
range $1/2<\sigma<1$.

######  Theorem 2.

Assume $\mu(1/2)<1/2k$. For every $\epsilon>0$

(1.26)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\int_{0}^{T}|\zeta(\sigma+it)|^{2k}\sin^{2}(k\theta_{Z_{N}}(\sigma+it))dt}{M_{k}(\sigma,T)}\left\\{\begin{array}[]{ll}=1/2\hskip
11.38092pt(N\in\mathbb{N},1/2<\sigma\leq\sigma_{k})\\\ <\epsilon\hskip
22.1931pt\left(N>N(\epsilon),\sigma_{k}<\sigma<1\right).\end{array}\right.$

One side of this law states that, if $\sigma_{k}>1/2$, then for every
$1/2<\sigma\leq\sigma_{k}$ and $N\in\mathbb{N}$ the distribution of the angles
$k\theta_{Z_{N}}(\sigma+it)$ $\pmod{(-\pi,\pi]}$ with respect to the density
(1.15) is consistent with that of a continuous random variable uniformly
distributed on $(-\pi,\pi]$ with respect to the density (1.15). On the other
hand, the other side of this law states that the expected value of those
angles with respect to the density (1.15) converges to $0$ as
$N\rightarrow\infty$ for every $\sigma_{k}<\sigma<1$. Thus we obtain a highly
structured consequence of the event $\sigma_{k}>1/2$ conditionally on
$\mu(1/2)<1/2k$. In light of Proposition 2 above, Theorem 2 lends support to
the hypothesis that it is more natural for the events $\mu(1/2)<1/2k$ and
$\sigma_{k}>1/2$ to be mutually exclusive, i.e. that
$\mu(1/2)<1/2k\Rightarrow\sigma_{k}=1/2$, which would be a favourable outcome
as it implies that $\sigma_{3}=1/2$.

The proofs of theorems 1 and 2 are postponed until Section 3 because several
details of the arguments are quite general and the functional analysis in
Section 2 handles them in a clearer way. The proofs of propositions 1 and 2
are given in Section 4.

##  2\. Preliminaries. Almost-periodicity

Throughout this paper we write $\zeta=\zeta(\sigma+it)$ for any fixed
$\sigma\in(1/2,1)$ unless otherwise specified, and $f,g\in
L_{2,\textrm{loc}}(\mathbb{R}^{+})$. Writing

(2.1) $\displaystyle\langle
f,g\rangle=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}f(t)\overline{g(t)}dt$

and $\|f\|=\sqrt{\langle f,f\rangle}$ when the limits exists, it is clear from
(1.14) and the reverse triangle inequality that if the $\|f_{N}\|$ exist then
$\|f\|=\lim_{N\rightarrow\infty}\|f_{N}\|$. For example, this is the case when
the $f_{N}$ in (1.14) are Bohr almost-periodic functions, that is, the $f_{N}$
have a (necessarily bounded and continuous) uniformly convergent generalised
Fourier series

(2.2) $\displaystyle g=\sum_{\lambda\in\mathbb{R}}c_{\lambda}e_{\lambda}$

where $e_{\lambda}(t)=e^{i\lambda t}$, in which case the Fourier coefficients
$c_{\lambda}=\langle g,e_{\lambda}\rangle$ are necessarily square-summable and
$\|f_{N}\|^{2}=\sum_{\lambda\in\mathbb{R}}|\langle
f_{N},e_{\lambda}\rangle|^{2}$. In this case, the $f$ in (1.14) belongs to the
closure of the space of Bohr functions in the seminorm $\|\cdot\|$. In other
words, the equivalence class $\phi+f$ for $\|\phi\|=0$ belongs to the Hilbert
space $B^{2}=B^{2}(\mathbb{R}^{+})$ of Besicovitch almost-periodic function on
$\mathbb{R}^{+}$, in which case we write $f_{N}\rightarrow f$ in $B^{2}$ as
$N\rightarrow\infty$. For $f,g\in B^{2}$ the inner product (2.1) exists and
the Parseval relation222The coefficients $\langle f,e_{\lambda}\rangle$ are
necessarily non-zero for at most countably many $\lambda\in\mathbb{R}$.

(2.3) $\displaystyle\langle f,g\rangle=\sum_{\lambda\in\mathbb{R}}\langle
f,e_{\lambda}\rangle\overline{\langle g,e_{\lambda}\rangle}$

holds. Conversely, if $f\in B^{2}$ then for instance we may take $f_{N}$ to be
a partial sum

(2.4) $\displaystyle f_{N}=\sum_{n\leq N}\langle f,e_{\lambda_{n}}\rangle
e_{\lambda_{n}}$

and verify (1.14) for this sequence using Bessel’s inequality. Thus, if $f\in
B^{2}$ then $f$ is not CNS. Reasoning along similar lines, we have the
following pointwise equivalence.

######  Proposition 3.

For each $k\in\mathbb{N}$, $\sigma_{k}=1/2\Leftrightarrow\zeta^{k}\in B^{2}$.

Therefore, since $\sigma_{1}=\sigma_{2}=1/2$, the functions
$\zeta,\zeta^{2}\in B^{2}$ are not CNS.

Although it is not known that $\sigma_{k}=1/2$ for $k>2$, our next proposition
permits us to compute the Fourier coefficient of $\zeta^{k}$ in any case.

######  Proposition 4.

For each $k\in\mathbb{N}$, if $\mu(1/2)<1/k$ then

(2.7)
$\displaystyle\langle\zeta^{k},e_{\lambda}\rangle=\left\\{\begin{array}[]{ll}\frac{d_{k}(n)}{n^{\sigma}}\hskip
35.56593pt(\lambda=-\log n\hskip 8.5359pt(n\in\mathbb{N}))\\\ 0\hskip
52.06859pt(\textrm{\emph{otherwise}}).\end{array}\right.$

Thus, by Proposition 4 and the fact that $\mu(1/2)<1/6$, there is a broader
class of “almost-periodic” functions to which the functions $\zeta^{k}$
($1\leq k\leq 6$) belong, which we now introduce.

######  Lemma 1.

Let $E=\\{e_{\lambda}:\lambda\in\mathbb{R}\\}$ and consider the set of all $f$
satisfying

$\displaystyle\|f\|_{A^{2}}^{2}=\sum_{\lambda\in\mathbb{R}}|\langle
f,e_{\lambda}\rangle|^{2}<\infty.$

Then the set $A^{2}=A^{2}(\mathbb{R}^{+})$ of equivalence classes $\phi+f$
with $\phi\in E^{\perp}$ is a Hilbert space with inner product

(2.8) $\displaystyle\langle
f,g\rangle_{A^{2}}=\sum_{\lambda\in\mathbb{R}}\langle
f,e_{\lambda}\rangle\overline{\langle g,e_{\lambda}\rangle}$

and we have

(2.9) $\displaystyle A^{2}=E^{\perp}\oplus B^{2}.$

It is clear from (2.9) that if $f\in A^{2}$ then $f\in
B^{2}\Leftrightarrow\|\phi\|=0$. However, the space $A^{2}$ does contain many
more functions than $B^{2}$ and it is easily checked that there exist $\phi\in
E^{\perp}$ for which $0<\|\phi\|\leq\infty$, for example

$\displaystyle\left\|\frac{\zeta\left(1/2+i\cdot\right)}{\log^{1/2}(\cdot+1)}\right\|=1\hskip
5.69046pt\textrm{ and }\hskip
5.69046pt\left\|\frac{\zeta\left(1/2+i\cdot\right)}{\log^{1/3}(\cdot+1)}\right\|=\infty.$

Nonetheless, if $f\in A^{2}$ has certain properties that we assume in
Proposition 5 below, then we may conclude that there are precisely two
possibilities: either $f\in B^{2}$, or $f$ is CNS. Before stating the result,
we note that $B^{2}$ has a particular group of unitary operators acting on it.
This is the content of Lemma 2.

######  Lemma 2.

The set $U=\\{u\in B^{2}:|u|=1\\}$ is a multiplicative subgroup of $B^{2}$. In
particular, if $u\in U$ then $u^{j}\in U$ $(j\in\mathbb{Z})$. Moreover, if
$u_{\lambda}=ue_{\lambda}$ then the set
$\\{u_{\lambda}:\lambda\in\mathbb{R}\\}$ is an orthonormal basis for $B^{2}$.
Equivalently, the map $f\rightarrow uf$ $(u\in U)$ is a unitary operator on
$B^{2}$.

Propositions 3, 4 and lemmas 1, 2 are proved in Section 5.

The main result of this section is Proposition 5, where we write
$\theta_{f}=\arg f$.

######  Proposition 5.

Let $g\in B^{2}$, $\langle f,e_{\lambda}\rangle=\langle g,e_{\lambda}\rangle$
and $\langle f^{2},e_{\lambda}\rangle=\langle g^{2},e_{\lambda}\rangle$
$(\lambda\in\mathbb{R})$, so that $f,f^{2}\in A^{2}$. Also let
$e^{i\theta_{f}},e^{i\theta_{g}}\in U$, $\langle
e^{i\theta_{f}},e_{\lambda}\rangle=\langle e^{i\theta_{g}},e_{\lambda}\rangle$
$(\lambda\in\mathbb{R})$, and let $e^{i\theta_{g_{N}}}$ be a sequence of Bohr
functions such that $e^{i\theta_{g_{N}}}\rightarrow e^{i\theta_{g}}$ in
$B^{2}$ as $N\rightarrow\infty$. Then we have the following.

  * (_a_)

$f\in B^{2}$ if and only if $f$ is not CNS.

  * (_b_)

If the maps $h\mapsto e^{i\theta_{f}}h$ and $h\mapsto e^{i\theta_{g}}h$ are
bounded operators on $A^{2}$, then $f\in B^{2}$.

###### Proof of Proposition 5.

As we have seen, if $f\in B^{2}$ then $f$ is not CNS, so we now assume that
$f$ is not CNS and prove the converse statement. Using the identity
$1-\cos(2x)=2\sin^{2}(x)$ we have

(2.10) $\displaystyle 0$ $\displaystyle\leq$
$\displaystyle\frac{1}{T}\int_{0}^{T}|f|^{2}dt-\Re\frac{1}{T}\int_{0}^{T}f^{2}e^{-2i\theta_{g_{N}}}dt$
$\displaystyle=$
$\displaystyle\frac{2}{T}\int_{0}^{T}|f|^{2}\sin^{2}(\theta_{f}-\theta_{g_{N}})dt$
$\displaystyle\leq$
$\displaystyle\frac{2\sup\\{x:\Pr(\sin^{2}(\theta_{f}-\theta_{g_{N}})\geq
x)>0\\}+2\epsilon}{T}\int_{0}^{T}|f|^{2}dt$

for every $\epsilon>0$ and $T\geq T(\epsilon)$ by (1.11). Since
$\lim_{N\rightarrow\infty}\|e^{i\theta_{f}}-e^{i\theta_{g_{N}}}\|=0$, there is
also an $N(\epsilon)$ such that
$\Pr(\sin^{2}(\theta_{f}-\theta_{g_{N}})\geq\epsilon)=0$ for $N\geq
N(\epsilon)$, so (3.2) shows that

$\displaystyle
0\leq\frac{1}{T}\int_{0}^{T}|f|^{2}dt-\Re\frac{1}{T}\int_{0}^{T}f^{2}e^{-2i\theta_{g_{N}}}dt\leq\frac{4\epsilon}{T}\int_{0}^{T}|f|^{2}dt$

for $N\geq N(\epsilon)$ and $T\geq T(\epsilon)$, which gives

(2.11)
$\displaystyle\Re\frac{1}{T}\int_{0}^{T}f^{2}e^{-2i\theta_{g_{N}}}dt\leq\frac{1}{T}\int_{0}^{T}|f|^{2}dt\leq\frac{1}{1-4\epsilon}\Re\frac{1}{T}\int_{0}^{T}f^{2}e^{-2i\theta_{g_{N}}}dt$

for $0<\epsilon<1/4$. Since $e^{-2i\theta_{g_{N}}}$ is a Bohr function it has
a uniformly convergent Fourier series of the form (2.2), so the map $f\mapsto
e^{-2i\theta_{g_{N}}}f$ is a bounded operator on $A^{2}$. Therefore, since
$f^{2}\in A^{2}$, the limits

(2.12) $\displaystyle\langle
f^{2},e^{2i\theta_{g_{N}}}\rangle=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}f^{2}e^{-2i\theta_{g_{N}}}dt$

exist and from (2.11) we see that the second moment $\|f\|^{2}$ exists and

(2.13) $\displaystyle\|f\|^{2}=\lim_{N\rightarrow\infty}\Re\langle
f^{2},e^{2i\theta_{g_{N}}}\rangle.$

Moreover, since $\langle f^{2},e_{\lambda}\rangle=\langle
g^{2},e_{\lambda}\rangle$ and $e^{2i\theta_{g_{N}}}$ is a Bohr function, we
have

(2.14) $\displaystyle\langle f^{2},e^{2i\theta_{g_{N}}}\rangle=\left\langle
g^{2},e^{2i\theta_{g_{N}}}\right\rangle=\left\langle
g,e^{2i\theta_{g_{N}}}\overline{g}\right\rangle=\left\langle
g,e^{2i(\theta_{g_{N}}-\theta_{g})}g\right\rangle.$

Since also

$\displaystyle\lim_{N\rightarrow\infty}\|e^{2i(\theta_{g_{N}}-\theta_{g})}-1\|=\lim_{N\rightarrow\infty}\|e^{2i\theta_{g_{N}}}-e^{2i\theta_{g}}\|=0$

and again because $e^{2i\theta_{g_{N}}}$ is a Bohr function, it follows that
$e^{2i(\theta_{g_{N}}-\theta_{g})}\in U\subset B^{2}$ so the map $h\mapsto
e^{2i(\theta_{g}-\theta_{g_{N}})}h$ is a bounded operator on $B^{2}$ by Lemma
2. Thus, since norm convergence implies weak convergence, by (2.13) and (2.14)
we have

(2.15) $\displaystyle\|f\|^{2}=\lim_{N\rightarrow\infty}\Re\left\langle
g,e^{2i(\theta_{g_{N}}-\theta_{g})}g\right\rangle=\langle
g,g\rangle=\|g\|^{2}.$

Lastly, writing

$\displaystyle g_{M}=\sum_{n\leq M}\langle g,e_{\lambda_{n}}\rangle
e_{\lambda_{n}}$

(not necessarily the same as $g_{N}$ above) and using (2.15) and $\langle
f,e_{\lambda}\rangle=\langle g,e_{\lambda}\rangle$, we have

$\displaystyle\|f-g_{M}\|^{2}$ $\displaystyle=$
$\displaystyle\|f\|^{2}+\|g_{M}\|^{2}-2\Re\langle f,\overline{g}_{M}\rangle$
$\displaystyle=$ $\displaystyle\|g\|^{2}-\|g_{M}\|^{2}$ $\displaystyle=$
$\displaystyle\|g-g_{M}\|^{2}\leq\epsilon\hskip 28.45274pt(M>M(\epsilon)),$

showing that $f\in B^{2}$.

To complete the proof we show that if the maps $h\mapsto e^{i\theta_{f}}h$ and
$h\mapsto e^{i\theta_{g}}h$ are also bounded operators on $A^{2}$ (and
therefore unitary) then $f\in B^{2}$. Indeed, we have

(2.16) $\displaystyle\|f\|^{2}=\langle|f|^{2},1\rangle_{A^{2}}=\langle
f^{2}e^{-2i\theta_{f}},1\rangle_{A^{2}}=\langle
f^{2},e^{2i\theta_{f}}\rangle_{A^{2}}$

for $f^{2}\in A^{2}$ because the map $h\mapsto e^{i\theta_{f}}h$ ($h\in
A^{2}$) is unitary. Then, from the definition of the $A^{2}$ inner product
(2.8), since $\langle f^{2},e_{\lambda}\rangle=\langle
g^{2},e_{\lambda}\rangle$, $e^{i\theta_{f}},e^{i\theta_{g}}\in U$ and $\langle
e^{i\theta_{f}},e_{\lambda}\rangle=\langle
e^{i\theta_{g}},e_{\lambda}\rangle$, (2.16) is equal to

$\displaystyle\langle g^{2},e^{2i\theta_{f}}\rangle_{A^{2}}=\langle
g^{2},e^{2i\theta_{g}}\rangle_{A^{2}}=\langle
g^{2}e^{-2i\theta_{g}},1\rangle_{A^{2}}=\langle|g|^{2},1\rangle_{A^{2}}=\|g\|^{2}$

which gives (2.15). ∎

##  3\. Proof of Theorems 1 and 2

###  3.1. Theorem 1

This follows from Proposition 5 if $f=\zeta^{k}$ satisfies the conditions of
Proposition 5 assuming that $\mu(1/2)<1/2k$. Firstly, this implies that

(3.3) $\displaystyle\langle
f^{j},e_{\lambda}\rangle=\left\\{\begin{array}[]{ll}\frac{d_{jk}(n)}{n^{\sigma}}\hskip
34.14322pt(\lambda=-\log n\hskip 8.5359pt(n\in\mathbb{N}))\\\ 0\hskip
54.06006pt(\textrm{otherwise}).\end{array}\right.$

for $j=1$ and $2$, by Proposition 4. Set $g_{N}=P_{N}^{k}$ and note that since
$B^{2}$ is a Hilbert space there is a $g\in B^{2}$ such that

(3.6) $\displaystyle\langle
g,e_{\lambda}\rangle=\lim_{N\rightarrow\infty}\langle
g_{N},e_{\lambda}\rangle=\left\\{\begin{array}[]{ll}\frac{d_{k}(n)}{n^{\sigma}}\hskip
34.14322pt(\lambda=-\log n\hskip 8.5359pt(n\in\mathbb{N}))\\\ 0\hskip
54.06006pt(\textrm{otherwise})\end{array}\right.$

and $\|g-g_{N}\|<\epsilon$ for $N>N(\epsilon)$. Also, by (2.3) we have

(3.7) $\displaystyle\langle g^{2},e_{\lambda}\rangle=\langle
g,\overline{g}e_{\lambda}\rangle$ $\displaystyle=$
$\displaystyle\sum_{\nu\in\mathbb{R}}\langle
g,e_{\nu}\rangle\overline{\langle\overline{g}e_{\lambda},e_{\nu}\rangle}$
$\displaystyle=$ $\displaystyle\sum_{\nu\in\mathbb{R}}\langle
g,e_{\nu}\rangle\langle g,e_{\lambda-\nu}\rangle$ $\displaystyle=$
$\displaystyle\sum_{m=1}^{\infty}\frac{d_{k}(m)}{m^{\sigma}}\langle
g,e_{\lambda+\log m}\rangle$

and, since $\lambda+\log m$ is the logarithm of the reciprocal of a natural
number if and only if $\lambda=-\log n$ and $m|n$, (3.7) is

(3.10) $\displaystyle\langle
g^{2},e_{\lambda}\rangle=\left\\{\begin{array}[]{ll}\sum_{m|n}\frac{d_{k}(m)}{m^{\sigma}}\frac{d_{k}(n/m)}{(n/m)^{\sigma}}=\frac{d_{2k}(n)}{n^{\sigma}}\hskip
42.67912pt(\lambda=-\log n\hskip 8.5359pt(n\in\mathbb{N}))\\\ 0\hskip
170.71652pt(\textrm{otherwise}).\end{array}\right.$

Thus by (3.3), (3.6) and (3.10), this $g\in B^{2}$ satisfies $\langle
f,e_{\lambda}\rangle=\langle g,e_{\lambda}\rangle$ and $\langle
f^{2},e_{\lambda}\rangle=\langle g^{2},e_{\lambda}\rangle$
$(\lambda\in\mathbb{R})$.

Now, since $\theta_{g_{N}}=\theta_{P^{k}_{N}}=\Im\log P^{k}_{N}$
$\pmod{(-\pi,\pi]}$ and

$\displaystyle\Im\log P^{k}_{N}(\sigma+it)=-k\sum_{2\leq n\leq
N^{2}}\frac{\Lambda_{N}(n)\sin\left(t\log n\right)}{n^{\sigma}\log
n}\pmod{(-\pi,\pi]}$

by (1.17), it follows that

$\displaystyle e^{i\theta_{g_{N}}(\sigma+it)}=\exp\left(-ik\sum_{2\leq n\leq
N^{2}}\frac{\Lambda_{N}(n)\sin\left(t\log n\right)}{n^{\sigma}\log n}\right)$

is a Bohr function because $e^{iy}$ $(y\in\mathbb{R})$ is uniformly continuous
and a uniformly continuous function of a uniformly convergent sequence is
uniformly convergent. Since also

(3.12)
$\displaystyle\lim_{N\rightarrow\infty}\left\|e^{i(\theta_{g_{N}}-\theta_{f})}-1\right\|=\lim_{N\rightarrow\infty}\left\|e^{i\theta_{g_{N}}}-e^{i\theta_{f}}\right\|=0$

by Proposition 2, it follows that $e^{i\theta_{f}}\in U$. Moreover, since

$\displaystyle\ell_{N}=\sum_{2\leq n\leq
N^{2}}\frac{\Lambda_{N}(n)}{n^{\sigma+i\cdot}\log n}$

is a Bohr function, we have $\ell_{N}\rightarrow\ell$ in $B^{2}$ as
$N\rightarrow\infty$ in which the limit $\ell\in B^{2}$ is necessarily bounded
except possibly on a null set, so that also the $\ell_{N}$ $(N\in\mathbb{N})$
are bounded except possibly on a null set. Therefore, since

$\displaystyle\left\|g-\exp\ell_{N}\right\|=\left\|g-g_{N}\right\|<\epsilon\hskip
28.45274pt(N>N(\epsilon)),$

we note that $g$ cannot vanish except possibly on a null set. As such, since

$\displaystyle\left\||g|-|g_{N}|e^{i(\theta_{g_{N}}-\theta_{g})}\right\|=\left\|g-g_{N}\right\|,$

it follows that

$\displaystyle\textrm{Pr}\left(\left|\theta_{g_{N}}-\theta_{g}\pmod{(-\pi,\pi]}\right|\geq\epsilon\right)=0\hskip
28.45274pt(N>N(\epsilon))$

so $\lim_{N\rightarrow\infty}\|e^{i\theta_{g_{N}}}-e^{i\theta_{g}}\|=0$. In
other words, $e^{i\theta_{g_{N}}}\rightarrow e^{i\theta_{g}}$ in $B^{2}$ as
$N\rightarrow\infty$ so $e^{i\theta_{g}}\in U$ and
$\|e^{i\theta_{f}}-e^{i\theta_{g}}\|=0$ by (3.12) and the triangle inequality.
Since the latter implies that $\langle
e^{i\theta_{f}},e_{\lambda}\rangle=\langle e^{i\theta_{g}},e_{\lambda}\rangle$
$(\lambda\in\mathbb{R})$ by Cauchy-Schwarz, we conclude that $f=\zeta^{k}$
satisfies the conditions of Proposition 5 provided that $\mu(1/2)<1/2k$.

###  3.2. Theorem 2

Noting the equivalence of (1.4) and (1.5), if $\sigma_{k}>1/2$ it must be that

(3.13)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{M_{k}(\sigma,T)}{T}=\infty\hskip
28.45274pt(1/2<\sigma\leq\sigma_{k}).$

Using the identity $1-\cos(2x)=2\sin^{2}(x)$, we then have

$\displaystyle
M_{k}(\sigma,T)-\Re\int_{0}^{T}\zeta^{2k}e^{-2ik\theta_{P_{N}}}dt=2\int_{0}^{T}|\zeta|^{2k}\sin^{2}(k\theta_{Z_{N}})dt$

so that

(3.14)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\int_{0}^{T}|\zeta|^{2k}\sin^{2}(k\theta_{Z_{N}})dt}{M_{k}(\sigma,T)}=\frac{1}{2}-\liminf_{T\rightarrow\infty}\frac{\Re\frac{1}{T}\int_{0}^{T}\zeta^{2k}e^{-2ik\theta_{P_{N}}}dt}{\frac{2}{T}M_{k}(\sigma,T)}.$

Since $\zeta^{2k}\in A^{2}$ if $\mu(1/2)<1/2k$, the numerator of the second
term on the right hand side of (3.14) converges to the limit
$\langle\zeta^{2k},e^{2ik\theta_{P_{N}}}\rangle$ as $T\rightarrow\infty$.
Therefore,

$\displaystyle\liminf_{T\rightarrow\infty}\frac{\Re\frac{1}{T}\int_{0}^{T}\zeta^{2k}e^{-2ik\theta_{P_{N}}}dt}{\frac{2}{T}M_{k}(\sigma,T)}$
$\displaystyle\sim$
$\displaystyle\liminf_{T\rightarrow\infty}\frac{\Re\langle\zeta^{2k},e^{2ik\theta_{P_{N}}}\rangle}{\frac{2}{T}M_{k}(\sigma,T)}$
$\displaystyle=$
$\displaystyle\frac{\Re\langle\zeta^{2k},e^{2ik\theta_{P_{N}}}\rangle}{2}\left(\limsup_{T\rightarrow\infty}\frac{M_{k}(\sigma,T)}{T}\right)^{-1}=0$

by (3.13).

On the other hand, if $\mu(1/2)<1/2k$, then a trivial modification of the
proof of Proposition 3 shows that $\zeta^{k}(\sigma+it)\in B^{2}$ for
$\sigma_{k}<\sigma<1$. In particular, $\zeta^{k}(\sigma+it)$ is not CNS for
$\sigma_{k}<\sigma<1$, in which case we have

$\displaystyle\limsup_{T\rightarrow\infty}\frac{\int_{0}^{T}|\zeta|^{2k}\sin^{2}(k\theta_{Z_{N}})dt}{M_{k}(\sigma,T)}\leq\sup\\{x:\Pr(\sin^{2}(k\theta_{Z_{N}})\geq
x)>0\\}<\epsilon\hskip 14.22636pt(N>N(\epsilon))$

by (1.11) and Proposition 2.

##  4\. Proof of Propositions 1 and 2

###  4.1. Proposition 1

Let $g$ be bounded, assume (1.12) and denote by $S^{c}$ the complement of $S$
in $\mathbb{R}^{+}$. Then for every null set $S$ we have

$\displaystyle\left|\int_{0}^{T}f^{2}gdt\right|$ $\displaystyle=$
$\displaystyle\left|\int_{S^{c}\cap[0,T]}f^{2}gdt+\int_{S\cap[0,T]}f^{2}gdt\right|$
$\displaystyle\leq$
$\displaystyle\left|\int_{S^{c}\cap[0,T]}f^{2}gdt\right|+o\left(\int_{0}^{T}|f|^{2}dt\right)$
$\displaystyle\leq$ $\displaystyle(1+o(1))\sup_{t\in
S^{c}\cap[0,T]}|g(t)|\int_{0}^{T}|f|^{2}dt.$

Since $S$ is an arbitrary null set, we minimise over those $S$ giving

(4.1)
$\displaystyle\limsup_{T\rightarrow\infty}\frac{\left|\int_{0}^{T}f^{2}gdt\right|}{\int_{0}^{T}|f|^{2}dt}\leq\inf_{|S|=0}\sup_{t\in
S^{c}}|g(t)|=\sup\\{x:\Pr(|g|\geq x)>0\\}$

which is (1.11). Conversely, assume (4.1) and take
$g=e^{-2i\theta_{f}}\chi_{S}$ where $\chi_{S}$ is the characteristic function
of $S$. Then

$\displaystyle\limsup_{T\rightarrow\infty}\frac{\int_{S\cap[0,T]}|f|^{2}dt}{\int_{0}^{T}|f|^{2}dt}=\limsup_{T\rightarrow\infty}\frac{\left|\int_{0}^{T}f^{2}gdt\right|}{\int_{0}^{T}|f|^{2}dt}\leq\sup\\{x:\Pr(|g|\geq
x)>0\\}=0$

which gives (1.12).

Now suppose that $\int_{0}^{T}|f|^{2}dt\gg T$. We have

$\displaystyle\left(\frac{\int_{S\cap[0,T]}|f|^{2}dt}{\int_{0}^{T}|f|^{2}dt}\right)^{1/2}$
$\displaystyle\sim$
$\displaystyle\left|\left(\frac{\int_{S\cap[0,T]}|f|^{2}dt}{\int_{0}^{T}|f|^{2}dt}\right)^{1/2}-\left(\frac{\int_{S\cap[0,T]}|f_{N,S}|^{2}dt}{\int_{0}^{T}|f|^{2}dt}\right)^{1/2}\right|$
$\displaystyle\leq$
$\displaystyle\left(\frac{\int_{S\cap[0,T]}|f-f_{N,S}|^{2}dt}{\int_{0}^{T}|f|^{2}dt}\right)^{1/2}$

so (1.13) implies (1.12). Conversely, (1.12) implies (1.13) by taking
$f_{N,S}=0$.

To complete the proof, we note that if $\int_{0}^{T}|f|^{2}dt\ll T$ then $f$
is bounded except possibly on a null set $S=S(f)$, so there is already a
bounded function $f_{N,S^{c}}$ such that $|f-f_{N,S^{c}}|^{2}<\epsilon$ on
$S^{c}$. Thus, taking

$\displaystyle f_{N}=f_{N,S}\chi_{S}+f_{N,S^{c}}\chi_{S^{c}},$

we have

$\displaystyle\frac{1}{T}\int_{0}^{T}|f-f_{N}|^{2}dt$ $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{S\cap[0,T]}|f-f_{N,S}|^{2}dt+\frac{1}{T}\int_{S^{c}\cap[0,T]}|f-f_{N,S^{c}}|^{2}dt$
$\displaystyle\ll$
$\displaystyle\frac{\int_{S\cap[0,T]}|f-f_{N,S}|^{2}dt}{\int_{0}^{T}|f|^{2}dt}+\frac{\epsilon}{T}\textrm{meas}\left(S^{c}\cap[0,T]\right)$

and using (1.13) gives (1.14). Conversely, (1.14) implies (1.13) if
$\int_{0}^{T}|f|^{2}dt\gg T$.

###  4.2. Proof of Proposition 2

First note that the union of all subsets of the interval $[0,T]$ that lie in a
$\delta$-neighbourhood of a zero of $\zeta(s)$ for some fixed
$0<\delta<|\sigma-1/2|$ is of size $T^{\alpha}$ for some $\alpha<1$ due to the
zero density bound (1.16), so we have $|t-\gamma|>\delta$ for any $t$ lying in
the remaining set. We now recall the formula (Gonek [6])

$\displaystyle\log Z_{N}(s)=\sum_{\rho}F_{2}\left((s-\rho)\log
N\right)-F_{2}\left((s-1)\log
N\right)+O\left(\frac{N^{-\sigma-2}}{(|t|+1)^{2}\log^{2}N}\right)$

where

$\displaystyle F_{2}(z)=\int_{z}^{\infty}\frac{e^{-2w}-e^{-w}}{w^{2}}dw$

and the sum is absolutely convergent for $s\neq\rho$. Since

(4.3) $\displaystyle\int_{(s-\rho)\log
N}^{\infty}\frac{e^{-w}dw}{w^{2}}=\int_{0}^{\infty}\frac{N^{-w-s+\rho}dw}{(w+s-\rho)^{2}\log
N},$

it is clear that when $\sigma>\beta$ the exponential integrals
$F_{2}\left((s-\rho)\log N\right)=o(1)$ as $N\rightarrow\infty$. Since the sum
over all terms is absolutely convergent, the sum over these terms is $o(1)$ as
$N\rightarrow\infty$, so it remains to consider those terms with
$\beta\geq\sigma$. For the latter, the integral over the interval
$w>\beta-\sigma$ in (4.3) is again $o(1)$ as $N\rightarrow\infty$, so we are
left with

(4.4) $\displaystyle\log Z_{N}(s)$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\rho=\beta+i\gamma\\\
\sigma\leq\beta\end{subarray}}\int_{0}^{\beta-\sigma}\frac{N^{2w+2i(\gamma-t)}-N^{w+i(\gamma-t)}}{(w-i(\gamma-t))^{2}}\frac{dw}{\log
N}+o(1)$ $\displaystyle\ll$ $\displaystyle\frac{N}{\log
N}\sum_{\begin{subarray}{c}\sigma\leq\beta\end{subarray}}\frac{1}{|\gamma-t|}\arctan\left(\frac{\beta-\sigma}{|\gamma-t|}\right)+\epsilon$

for $N>N(\epsilon)$.

Since $\arctan(y)\ll y$ as $y\rightarrow 0$ and $\arctan(y)\leq 1$ as
$y\rightarrow\infty$, let $\Delta=\Delta(T)\rightarrow\infty$ as
$T\rightarrow\infty$ and split the sum in (4.4) depending on whether
$|\gamma-t|\leq\Delta$ or $|\gamma-t|>\Delta$, so that (4.4) is

(4.5) $\displaystyle\ll\frac{N}{\log
N}\sum_{\begin{subarray}{c}\sigma\leq\beta\\\
|\gamma-t|\leq\Delta\end{subarray}}\frac{1}{|\gamma-t|}+\frac{N}{\log
N}\sum_{\begin{subarray}{c}\sigma\leq\beta\\\
|\gamma-t|>\Delta\end{subarray}}\frac{1}{(\gamma-t)^{2}}+\epsilon$

as $T\rightarrow\infty$. To handle the second sum in (4.5), we note that it is
less than

$\displaystyle\sum_{\begin{subarray}{c}|\gamma-t|>\Delta\end{subarray}}\frac{1}{(\gamma-t)^{2}}\ll\frac{\log\Delta}{\Delta}+\frac{\log
t}{\Delta}$

due to Gonek [6]. Taking $\Delta=T^{A}$ for any $A>0$, we have

(4.6) $\displaystyle\log Z_{N}(s)\ll\frac{N}{\log
N}\sum_{\begin{subarray}{c}\sigma\leq\beta\\\ |\gamma-t|\leq
T^{A}\end{subarray}}\frac{1}{|\gamma-t|}+\frac{N}{T^{A}\log N}+\epsilon$

as $T\rightarrow\infty$.

Lastly, observe that the sum in (4.6) is empty unless $|\gamma-t|\leq T^{A}$
for at least one $\gamma$ and, since the total number of zeros with
$\sigma\leq\beta$ is $\ll T^{\alpha}$ for some fixed $\alpha<1$, the union of
all such subsets is of size $\ll T^{\alpha+A}$. Therefore, taking
$A=1-\alpha-\delta$ for sufficiently small $\delta$, the union of all such
subsets is of size $\ll T^{1-\delta}$. So for all fixed $1/2<\sigma<1$ and $t$
lying in the remaining set we have

(4.7) $\displaystyle\log Z_{N}(s)\ll\epsilon$

for $N(\epsilon)<N\ll T^{1-\alpha-\delta}$ and $T\rightarrow\infty$. Since
$\Im\log Z_{N}=\theta_{Z_{N}}$ $\pmod{(-\pi,\pi]}$, this completes the proof.

##  5\. proofs of the auxiliary results

###  5.1. Proposition 3

Clearly $\zeta^{k}\in B^{2}$ implies that $\sigma_{k}=1/2$, so we assume
$\sigma_{k}=1/2$ and prove the converse statement. It follows from
$\mu(1/2)<1/2k$ and Proposition 4 that

(5.3)
$\displaystyle\langle\zeta^{k},e_{\lambda}\rangle=\left\\{\begin{array}[]{ll}\frac{d_{k}(n)}{n^{\sigma}}\hskip
35.56593pt(\lambda=-\log n\hskip 8.5359pt(n\in\mathbb{N}))\\\ 0\hskip
52.06859pt(\textrm{otherwise})\end{array}\right.$

so, if we set $f=\zeta^{k}$ and

$\displaystyle f_{N}=\sum_{n\leq N}\frac{d_{k}(n)}{n^{\sigma}}n^{-i\cdot},$

then

(5.4) $\displaystyle\|f-f_{N}\|^{2}=\|f\|^{2}+\|f_{N}\|^{2}-2\Re\langle
f,\overline{f}_{N}\rangle=\|f\|^{2}-\|f_{N}\|^{2}=\sum_{n>N}\frac{d^{2}_{k}(n)}{n^{2\sigma}}$

by (1.6). Since (5.4) is clearly $<\epsilon$ for $N>N(\epsilon)$, we see that
$\sigma_{k}=1/2\Rightarrow\zeta^{k}\in B^{2}$.

###  5.2. Proposition 4

Fix $\delta>0$. Denote by $R$ the rectangle with vertices
$[\sigma+iT,1+\delta+iT,1+\delta+i,\sigma+i]$ and consider the integral

(5.5)
$\displaystyle\frac{1}{2iT}\int_{R}\zeta^{k}(s)e^{\lambda(2\sigma-s)}ds.$

Assuming that $\mu(1/2)<1/k$, there is an $A>0$ such that the sum of the
horizontal segments is

(5.6)
$\displaystyle\ll\frac{1}{T}\int_{\sigma}^{1+\delta}|\zeta(r+iT)|^{k}e^{\lambda(2\sigma-r)}dr+\frac{e^{\lambda\sigma}}{T}\ll\frac{e^{\lambda\sigma}}{T^{A}}$

and the sum of the vertical segments is

(5.7)
$\displaystyle\frac{e^{2\lambda\sigma}}{T}\int_{1}^{T}\zeta^{k}(1+\delta+it)e^{-\lambda(1+\delta+it)}dt-\frac{e^{\lambda\sigma}}{T}\int_{1}^{T}\zeta^{k}(\sigma+it)e^{-i\lambda
t}dt$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{d_{k}(n)}{n^{1+\delta}}\frac{e^{\lambda(2\sigma-1-\delta)}}{T}\int_{1}^{T}e^{-i(\lambda+\log
n)t}dt-\frac{e^{\lambda\sigma}}{T}\int_{1}^{T}\zeta^{k}(\sigma+it)e^{-i\lambda
t}dt$

because the Dirichlet series above is absolutely convergent. By (5.5), (5.6)
and (5.7) we see that

(5.10) $\displaystyle\frac{1}{T}\int_{1}^{T}\zeta^{k}(\sigma+it)e^{-i\lambda
t}dt$ $\displaystyle=$ $\displaystyle
e^{\lambda(\sigma-1-\delta)}\sum_{n=1}^{\infty}\frac{d_{k}(n)}{n^{1+\delta}}\frac{1}{T}\int_{1}^{T}e^{-i(\lambda+\log
n)t}dt+O(T^{-A})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{d_{k}(n)}{n^{\sigma}}+o\left(1\right)\hskip
35.56593pt(\lambda=-\log n)\\\ o\left(1\right)\hskip
72.55461pt(\textrm{otherwise}).\end{array}\right.$

###  5.3. Lemma 1

$A^{2}$ is an inner product space by definition. Since $B^{2}$ is a dense
subset (in the $A^{2}$ metric, of course), the completeness of $A^{2}$ is
inherited from that of $B^{2}$ and for every $f\in A^{2}$ we then have
$f=\phi+g$ pointwise for some $\phi\in E^{\perp}$ and $g\in B^{2}$.

###  5.4. Lemma 2

Let $u\in B^{2}$ and $\nu\in\mathbb{R}$ be fixed. Since the set
$\\{e_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is a complete orthonormal set in
$B^{2}$, for every $\epsilon>0$ there is an integer $M(u,\epsilon)$ and a
sequence of real numbers $\lambda_{1},\cdots,\lambda_{M}$ such that

(5.11) $\displaystyle\left\|\overline{u}-\sum_{m\leq
M}\left\langle\overline{u},e_{\lambda_{m}-\nu}\right\rangle
e_{\lambda_{m}-\nu}\right\|<\epsilon\hskip 28.45274pt(M>M(u,\epsilon)).$

If $|u|=1$, then it is immediate from (5.11) that

(5.12) $\displaystyle\left\|e_{\nu}-\sum_{m\leq M}\left\langle
e_{\nu},ue_{\lambda_{m}}\right\rangle ue_{\lambda_{m}}\right\|<\epsilon\hskip
28.45274pt(M>M(u,\epsilon)).$

For each $\lambda\in\mathbb{R}$ we write $u_{\lambda}=ue_{\lambda}$ and note
that the set $\\{u_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is an orthonormal set
in $B^{2}$. Now suppose that the set
$\\{u_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is not dense in $B^{2}$, and let
$V$ be the proper subspace spanned by
$\\{u_{\lambda}\\}_{\lambda\in\mathbb{R}}$. Since
$\\{e_{\nu}\\}_{\nu\in\mathbb{R}}$ is a complete orthonormal set there is an
$e_{\nu_{0}}\in V^{\perp}$, which contradicts (5.12) so
$\\{u_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is dense in $B^{2}$. Since a dense
orthonormal set in a Hilbert space is complete, it follows that the set
$\\{u_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is a complete orthonormal set in
$B^{2}$ and by Parseval’s theorem we have

$\displaystyle\sum_{\lambda\in\mathbb{R}}|\langle\overline{u}f,e_{\lambda}\rangle|^{2}=\sum_{\lambda\in\mathbb{R}}|\langle
f,u_{\lambda}\rangle|^{2}=\|f\|^{2}\hskip 28.45274pt(f\in B^{2}).$

Thus the transformations $M_{u}$ are continuous on $B^{2}$ and therefore
unitary.

Lastly, to see that $U$ is a multiplicative subgroup, note that if $u,v\in U$
and $uv=w$ then

$\displaystyle\left\|w-\sum_{m\leq M}\left\langle
w,v_{\lambda_{m}}\right\rangle v_{\lambda_{m}}\right\|=\left\|u-\sum_{m\leq
M}\left\langle u,e_{\lambda_{m}}\right\rangle
e_{\lambda_{m}}\right\|<\epsilon$

which implies $w\in U$ because $\\{v_{\lambda}\\}_{\lambda\in\mathbb{R}}$ is a
complete orthonormal set in $B^{2}$.

#### Acknowledgement

I would like to thank Julio Andrade, Roger Heath-Brown and Christopher Hughes
for their comments and suggestions on this work. I am also grateful to be
supported by the Faculty of Environment, Science and Economy at the University
of Exeter.

## References

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# Casimir-Lifshitz force for graphene-covered gratings

Youssef Jeyar<EMAIL_ADDRESS>Laboratoire Charles Coulomb
(L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier, France
Minggang Luo  Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de
Montpellier, F-34095 Montpellier, France  Brahim Guizal  Laboratoire Charles
Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, F-34095 Montpellier,
France  H. B. Chan  Department of Physics, The Hong Kong University of Science
and Technology, Clear Water Bay, Kowloon, Hong Kong, China  William Mong
Institute of Nano Science and Technology, The Hong Kong University of Science
and Technology, Clear Water Bay, Kowloon, Hong Kong, China  Center for
Metamaterial Research, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, China  Mauro Antezza
<EMAIL_ADDRESS>Laboratoire Charles Coulomb (L2C), UMR 5221
CNRS-Université de Montpellier, F-34095 Montpellier, France  Institut
Universitaire de France, 1 rue Descartes, Paris Cedex 05, F-75231, France

###### Abstract

We study the Casimir-Lifshitz force (CLF) between a gold plate and a graphene-
covered dielectric grating. Using a scattering matrix (S-matrix) approach
derived from the Fourier Modal Method (FMM), we find a significant enhancement
in the CLF as compared to a mere dielectric slab coated with graphene, over a
wide range of temperatures. Additionally, we demonstrate that the CLF depends
strongly on the chemical potential of graphene, with maximal effects observed
at lower filling fractions. Finally, we analyse the Casimir force gradient
between a gold sphere and a graphene-coated dielectric grating, highlighting
potential avenues for experimental measurements.

††preprint: APS/123-QED

##  I Introduction

The Casimir-Lifshitz force (CLF), resulting from quantum vacuum fluctuations,
manifests in the interactions between polarizable bodies. In 1948, Casimir [1]
first developed a theoretical framework for this force between two perfectly
conducting plates at zero temperature. Later, Lifshitz expanded this framework
to encompass bodies with diverse optical properties and at finite temperatures
[2]. Since its experimental verification [3], the CLF has attracted
significant research interest, with investigations focused on exploring
different geometries and materials [4, 5, 6, 7, 8, 9, 10, 11].

Among diverse geometrical systems, the grating configuration is particularly
important in the study of the CLF and has been explored both theoretically
[12, 13] and experimentally [14, 15], in terms of the non-trivial interplay
between geometry and dielectric properties of the interacting bodies.
Analyzing such structures necessitates precise and efficient methods such as
the Fourier Modal Method (FMM) or its improved version, the FMM with adaptive
spacial resolution (FMM-ASR), which significantly enhances efficiency for
metallic gratings [16].

The dielectric properties of the interacting objects can affect the scattering
details, which are relevant to both CLF in and out of thermal equilibrium and
radiative heat transfer. In particular, there has been much progress in
graphene-based structures [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28],
that utilizes the unique optical properties of this material [29, 30, 31, 32,
33]. A recent study [34] delved into the CLF between graphene gratings,
utilizing another enhanced version of the FMM approach known as FMM with local
basis functions [35]. This study shows the potential for modulating the force
by manipulating geometry parameters and/or graphene properties. The
investigation is further extended to analyzing heat transfer within these
structures for both aligned [36] and misaligned [37] cases. Patterning the
graphene sheet into grating will result in a topology transition of the modes
for energy transfer from the circular one to the hyperbolic one, which allows
a significant enhancement in the heat transfer.

In another example [38], graphene is integrated with one-dimensional plasmonic
nanogratings with high aspect ratios and narrow grooves, the good agreement
between theory and experiment suggests promising prospects for high-
performance electrically tunable graphene-based infrared photodetectors and
biological sensors.

In this work, we explore the CLF between graphene coated dielectric grating
and gold semi-finite plane, using the scattering matrix approach obtained by
an adapted FMM to include the graphene sheet. This configuration combines the
features of both the grating structures and the special dielectric properties
of graphene, which is expected to provide more freedom in controlling the CLF.

In section II, we describe the physical system, in section III we introduce
the reflection operators obtained by the FMM. Finally, in section IV, we
present and discuss the numerical results. The calculation details of the
scattering matrix are provided in the appendix.

Figure 1: Schematic of the system: a dielectric grating covered with graphene
sheet and is placed at a distance $d$ from a gold half-space. The grating is
periodic in the $x$ direction with period $D$, characterized by width $a$, a
depth $h_{1}$ and filling fraction $f=a/D$.

##  II Physical system

The physical system depicted in Fig. 1 consists of a dielectric grating coated
with a graphene sheet (body 1) facing a gold half-space (body 2). These two
bodies are separated by a distance $d$ and maintained at a temperature $T$.
The dielectric grating, composed of fused silica (SiO2), exhibits periodicity
in the $x$ direction with a period $D$ (we use a fixed value of $D=1\mu$m in
this study), each period contains an SiO2 region of width $a$, giving a
filling fraction $f=a/D$. The grating has a thickness $h_{1}$ of 300 nm and is
positioned on top of an SiO2 slab with a thickness $h_{2}$ of 100 nm, which in
turn sits on an silicon (Si) half-space.

For this configuration, the Casimir-Lifshitz pressure (CLP) acting on body 1
along the $z$ positive direction can be expressed as [13, 39, 34]

$P(d,T,\mu)=\frac{k_{\rm
B}T}{4\pi^{2}}\sum_{m=0}^{+\infty}{}^{\prime}\int_{-\frac{\pi}{D}}^{\frac{\pi}{D}}{\rm
d}k_{x}\int_{-\infty}^{+\infty}{\rm d}k_{y}{\rm
Tr}\left(\gamma^{\prime}\mathcal{M}\right),$ (1)

where $\gamma^{\prime}={\rm diag}(k_{zn}^{\prime},k_{zn}^{\prime})_{n}$,
$k_{zn}^{\prime}=\sqrt{\xi_{m}^{2}/c^{2}+{\bf{k}}_{n}^{2}}$,
${\bf{k}}_{n}=(k_{xn},k_{y})$, $k_{xn}=k_{x}+n\frac{2\pi}{D}$, $k_{x}$ is in
the first Brillouin zone $[-\frac{\pi}{D},\frac{\pi}{D}]$ and $k_{y}$ is in
$\mathbb{R}$. The notation $\operatorname{diag}\left(u_{n},u_{n}\right)_{n}$
represents a diagonal matrix of size $2(2N+1)\times 2(2N+1)$, where the
diagonal elements are
$u_{-N},u_{-N+1},\ldots,u_{N},u_{-N},u_{-N+1},\ldots,u_{N}$. The expression
for $\mathcal{M}$ is given by

$\mathcal{M}=(U^{(12)}\mathcal{R}^{(1)+}\mathcal{R}^{(2)-}+U^{(21)}\mathcal{R}^{(2)-}\mathcal{R}^{(1)+}).$
(2)

where $\mathcal{R}^{(1)+}$ and $\mathcal{R}^{(2)-}$ represent the reflection
operators corresponding to the two bodies in the (TE, TM) basis obtained using
the scattering matrix approach, as will be discussed hereafter. As for the
inter-cavity operators, they are defined as follows ($\mathcal{I}$ denoting
the operator identity):

$\displaystyle U^{(12)}$
$\displaystyle=(\mathcal{I}-\mathcal{R}^{(1)+}\mathcal{R}^{(2)-})^{-1},$ (3)
$\displaystyle U^{(21)}$
$\displaystyle=(\mathcal{I}-\mathcal{R}^{(2)-}\mathcal{R}^{(1)+})^{-1}.$

In Eq. (1), the sum is over the Matsubara frequencies ($\xi_{m}=2\pi mk_{\rm
B}T/\hbar$) and the prime on this sum means that the first term ($m$=0) is
devided by 2. $k_{\rm{B}}$ is the Boltzmann constant, and $\hbar$ is the
reduced Planck’s constant.

For fused silica, the dielectric response at imaginary frequencies, is
obtained through the Kramers-Kronig relation :
$\varepsilon(i\xi_{m})=1+2\pi^{-1}\int_{0}^{\infty}\omega\varepsilon^{\prime\prime}(\omega)/(\omega^{2}+\xi_{m}^{2})d\omega$
and relies on the data extracted from the dielectric function of SiO2, as
sourced from [40], extrapolated along the real frequency axis.

We adopt the following model for the dielectric function of silicon [15]:

$\varepsilon_{\rm
Si}(i\xi_{m})=1.035+\frac{11.87-1.035}{1+\xi_{m}^{2}/\omega_{0}^{2}}+\frac{\omega_{p}^{2}}{\xi_{m}(\xi_{m}+\Gamma)}$
(4)

where $\omega_{0}=6.6\times 10^{15}$ rad s-1, $\omega_{p}=2.37\times 10^{14}$
rad s-1, and $\Gamma=6.45\times 10^{13}$ rad s-1.

We employ the Drude permittivity model for the dielectric function of gold
(Au)

$\varepsilon_{\rm
Au}(i\xi_{m})=1+\frac{\omega_{p}^{{}^{\prime}2}}{\xi_{m}(\xi_{m}+\gamma)}$ (5)

with $\hbar\omega_{p}^{{}^{\prime}}$= 9 eV and $\hbar\gamma$= 35 meV.

The contribution of graphene is introduced through its conductivity, which
varies with temperature $T$ and chemical potential $\mu$. We model the
conductivity as the sum of intraband and interband components: $\sigma_{\rm
g}=\sigma_{\textnormal{intra}}+\sigma_{\textnormal{inter}}$. More precisely,
their expressions on the imaginary frequency axis are given by [26,29,30,31]

$\displaystyle\sigma_{\textnormal{intra}}(i\xi_{m})$
$\displaystyle=\dfrac{8\sigma_{0}k_{B}T}{\pi(\hbar\xi_{m}+\hbar/\tau)}\ln\left[2\cosh\left(\dfrac{\mu}{2k_{B}T}\right)\right],$
(6) $\displaystyle\sigma_{\textnormal{inter}}(i\xi_{m})$
$\displaystyle=\dfrac{\sigma_{0}4\hbar\xi_{m}}{\pi}\int_{0}^{+\infty}\dfrac{G(x)}{(\hbar\xi_{m})^{2}+4x^{2}}dx,$

where, $\sigma_{0}={e^{2}}/({4\hbar})$, $e$ is the electron charge,
$G(x)=\sinh(x/k_{B}T)/[\cosh(\mu/k_{B}T)+\cosh(x/k_{B}T)]$, $\tau$ the
relaxation time (we use $\tau=10^{-13}$s).

##  III Reflection operators

In this section, we present the derivation of the reflection operators for the
interacting bodies. We begin with body 1, which comprises a dielectric grating
covered with graphene, as depicted in Fig. 1. The reflection operator can be
directly obtained, as sub-blocs, from the scattering matrix, which has
dimensions of $4(2N+1)\times 4(2N+1)$ and can represented as follows

$\mathbb{S}=\begin{pmatrix}\mathcal{R}^{-}&\mathcal{T}^{-}\\\
\mathcal{T}^{+}&\mathcal{R}^{+}\end{pmatrix}.$ (7)

This matrix is obtained through the use of the FMM (see detailed calculations
in the appendix) and takes the form:

$\mathbb{S}=\left(\begin{array}[]{cc}\mathbb{1}&\mathbb{0}\\\
\mathbb{0}&\Phi\end{array}\right)\mathbb{\tilde{S}}\left(\begin{array}[]{cc}\mathbb{1}&\mathbb{0}\\\
\mathbb{0}&\Phi\end{array}\right),$ (8)

where

$\displaystyle\Phi\equiv$
$\displaystyle\left(\begin{array}[]{cc}\operatorname{diag}\left(e^{ik_{zn}^{(4)}(h_{1}+h_{2})}\right)_{n}&\mathbb{0}\\\
\mathbb{0}&\operatorname{diag}\left(e^{ik_{zn}^{(4)}(h_{1}+h_{2})}\right)_{n}\end{array}\right),$
(9)

$k_{zn}^{(i)}=\sqrt{\varepsilon_{i}(\omega)\dfrac{\omega^{2}}{c^{2}}-{\bf{k}}_{n}^{2}}$
for medium $i$ (see details in the appendix), and

$\mathbb{\tilde{S}}=\mathbb{\tilde{S}}_{1}\circledast\mathbb{\tilde{S}}_{2}\circledast\mathbb{\tilde{S}}_{3}.$
(10)

Here, the star product operation
$\mathbbm{A}=\mathbbm{B}\circledast\mathbbm{C}$ is defined as [16]

$\displaystyle\mathbbm{A}_{11}=\mathbbm{B}_{11}+\mathbbm{B}_{12}(\mathbbm{1}-\mathbbm{C}_{11}\mathbbm{B}_{22})^{-1}\mathbbm{C}_{11}\mathbbm{B}_{21},$
(11)
$\displaystyle\mathbbm{A}_{12}=\mathbbm{B}_{12}(\mathbbm{1}-\mathbbm{C}_{11}\mathbbm{B}_{22})^{-1}\mathbbm{C}_{12},$
$\displaystyle\mathbbm{A}_{21}=\mathbbm{C}_{21}(\mathbbm{1}-\mathbbm{B}_{22}\mathbbm{C}_{11})^{-1}\mathbbm{B}_{21},$
$\displaystyle\mathbbm{A}_{22}=\mathbbm{C}_{22}+\mathbbm{C}_{21}(\mathbbm{1}-\mathbbm{B}_{22}\mathbbm{C}_{11})^{-1}\mathbbm{B}_{22}\mathbbm{C}_{12},$

and

$\displaystyle\mathbb{\tilde{S}}_{1}$
$\displaystyle=\left(\begin{array}[]{ll}\;\;\;\;\mathbb{K}_{1}^{\prime}&-\mathbb{P}\\\
\mathbb{L}_{1}^{\prime}+\mathbb{M}_{1}^{\prime}&-\mathbb{P}^{\prime}\end{array}\right)^{-1}\left(\begin{array}[]{cc}\mathbb{K}_{1}&\mathbb{P}\\\
\mathbb{L}_{1}+\mathbb{M}_{1}&-\mathbb{P}^{\prime}\end{array}\right),\;\;\;\;\;\;\;\;\;\;\;\;$
(12) $\displaystyle\mathbb{\tilde{S}}_{2}$
$\displaystyle=\left(\begin{array}[]{cc}\Phi_{1}&\mathbb{0}\\\
\mathbb{0}&\mathbb{1}\end{array}\right)\left(\begin{array}[]{ll}-\mathbb{P}&-\mathbb{K}_{3}\\\
\mathbb{P}^{\prime}&-\mathbb{L}_{3}\end{array}\right)^{-1}\left(\begin{array}[]{ll}\mathbb{P}&-\mathbb{K}_{3}^{\prime}\\\
\mathbb{P}^{\prime}&-\mathbb{L}_{3}^{\prime}\end{array}\right)\left(\begin{array}[]{cc}\Phi_{1}&\mathbb{0}\\\
\mathbb{0}&\mathbb{1}\end{array}\right),$
$\displaystyle\mathbb{\tilde{S}}_{3}$
$\displaystyle=\left(\begin{array}[]{cc}\Phi_{2}&\mathbb{0}\\\
\mathbb{0}&\mathbb{1}\end{array}\right)\left(\begin{array}[]{ll}\mathbb{K}_{3}^{\prime}&\mathbb{K}_{4}\\\
\mathbb{L}_{3}^{\prime}&\mathbb{L}_{4}\end{array}\right)^{-1}\left(\begin{array}[]{ll}\mathbb{K}_{3}&\mathbb{K}_{4}^{\prime}\\\
\mathbb{L}_{3}&\mathbb{L}_{4}^{\prime}\end{array}\right)\left(\begin{array}[]{cc}\Phi_{2}&\mathbb{0}\\\
\mathbb{0}&\mathbb{1}\end{array}\right).$

In these matrices, we have introduced

$\displaystyle\quad\mathbb{L}_{i}$
$\displaystyle=\sqrt{\varepsilon_{i}}\left(\begin{array}[]{cc}\mathbb{B}_{xi}&\mathbb{A}_{y}\\\
\mathbb{B}_{yi}&-\mathbb{A}_{x}\end{array}\right),\quad\;\;\mathbb{L}_{i}^{\prime}=\sqrt{\varepsilon_{i}}\left(\begin{array}[]{cc}\mathbb{B}_{xi}&-\mathbb{A}_{y}\\\
\mathbb{B}_{yi}&\mathbb{A}_{x}\end{array}\right),$ (13)
$\displaystyle\mathbb{K}_{i}$
$\displaystyle=\left(\begin{array}[]{cc}\mathbb{A}_{y}&-\mathbb{B}_{xi}\\\
-\mathbb{A}_{x}&-\mathbb{B}_{yi}\end{array}\right),\;\;\;\;\;\;\;\;\;\mathbb{K}_{i}^{\prime}=\left(\begin{array}[]{cc}-\mathbb{A}_{y}&-\mathbb{B}_{xi}\\\
\mathbb{A}_{x}&-\mathbb{B}_{yi}\end{array}\right),\;\;\;\;\;\;\;$
$\displaystyle\mathbb{M}_{i}$ $\displaystyle=\small{\sigma_{\rm
g}Z_{0}}\left(\begin{array}[]{cc}-\mathbb{A}_{x}&-\mathbb{B}_{yi}\\\
-\mathbb{A}_{y}&\mathbb{B}_{xi}\end{array}\right),\;\;\mathbb{M^{\prime}}_{i}=\small{\sigma_{\rm
g}Z_{0}}\left(\begin{array}[]{cc}\mathbb{A}_{x}&-\mathbb{B}_{yi}\\\
\mathbb{A}_{y}&\mathbb{B}_{xi}\end{array}\right),$

with

$\displaystyle\;\;\;\;\;\;\;\mathbb{A}_{x}$
$\displaystyle=\operatorname{diag}\left(\frac{k_{xn}}{k_{n}}\right)_{n},\quad\mathbb{A}_{y}=\operatorname{diag}\left(\frac{k_{y}}{k_{n}}\right)_{n},$
(14) $\displaystyle\mathbb{B}_{xi}$
$\displaystyle=\frac{c}{\sqrt{\varepsilon_{i}}\omega}\operatorname{diag}\left(\frac{k_{xn}}{k_{n}}k_{zn}^{(i)}\right)_{n},$
$\displaystyle\mathbb{B}_{yi}$
$\displaystyle=\frac{c}{\sqrt{\varepsilon_{i}}\omega}\operatorname{diag}\left(\frac{k_{y}}{k_{n}}k_{zn}^{(i)}\right)_{n},$

and

$\displaystyle\;\;\;\;\;\Phi_{1}\equiv$ $\displaystyle
e^{\mathbb{D}h_{1}}=\left(\begin{array}[]{cc}e^{\mathbb{D}^{(11)}h_{1}}&\mathbb{0}\\\
\mathbb{0}&e^{\mathbb{D}^{(22)}h_{1}}\end{array}\right),$ (15)
$\displaystyle\Phi_{2}\equiv$
$\displaystyle\left(\begin{array}[]{cc}\operatorname{diag}\left(e^{ik_{zn}^{(3)}h_{2}}\right)_{n}&\mathbb{0}\\\
\mathbb{0}&\operatorname{diag}\left(e^{ik_{zn}^{(3)}h_{2}}\right)_{n}\end{array}\right).$

$Z_{0}$ denotes the impedance of vacuum.

The reflection matrix of body 1, denoted as $\mathcal{R}^{(1)+}$, can be
derived directly from the matrix $\mathcal{R}^{-}$. Specifically, due to the
distinct orientations along the z-axis, $\mathcal{R}^{(1)+}$ shares identical
characteristics with $\mathcal{R}^{-}$ within the two diagonal blocks.
However, a sign difference is observed within the two off-diagonal blocks,
detailed in [16], as follows

$\mathcal{R}^{(1)+}=\begin{cases}\mathcal{R}^{-}_{p,p^{\prime}}&p=p^{\prime}\\\
-\mathcal{R}^{-}_{p,p^{\prime}}&p\neq p^{\prime}.\end{cases}$ (16)

For body 2, positioned at a distance $d$ from the origin, a phase shift is
induced in the reflection matrix, as outlined in [39]. This reflection matrix
can be formulated as

$\displaystyle\left\langle
p,{\textbf{k}},n|\mathcal{R}^{(2)-}(\omega)|p^{\prime},{\textbf{k}}^{\prime},n^{\prime}\right\rangle$
(17) $\displaystyle=e^{i\left(k_{zn}+k_{zn^{\prime}}^{\prime}\right)d}$
$\displaystyle\left\langle
p,{\textbf{k}},n|\mathcal{\tilde{R}}^{(2)-}(\omega)|p^{\prime},{\textbf{k}}^{\prime},n^{\prime}\right\rangle.$

Here, $\mathcal{\tilde{R}}^{(2)-}$ is diagonal with dimensions $2(2N+1)\times
2(2N+1)$ and can be easily obtained using the Fresnel coefficients:

$\mathcal{\tilde{R}}^{(2)-}={\rm diag}(\rho_{n}^{\rm TE},\rho_{n}^{\rm TM})$
(18)

where

$\displaystyle\rho_{n}^{\rm TE}$ $\displaystyle=\dfrac{k_{zn}-k_{zn}^{\rm
Au}}{k_{zn}+k_{zn}^{\rm Au}}$ (19) $\displaystyle\rho_{n}^{\rm TM}$
$\displaystyle=\dfrac{\varepsilon_{\rm Au}(\omega)k_{zn}-k_{zn}^{\rm
Au}}{\varepsilon_{\rm Au}(\omega)k_{z}+k_{z}^{\rm Au}}$ (20)

In these expressions,
$k_{zn}=\sqrt{\dfrac{\omega^{2}}{c^{2}}-{\bf{k}}_{n}^{2}}$.

Finally, it is worth noting that this calculation is also valid for imaginary
Matsubara frequencies by replacing $\omega$ with $i\xi_{m}$.

##  IV Results and discussion

###  IV.1 The effect of graphene coating

Now that we have all the necessary ingredients to calculate the Casimir-
Lifshitz pressure for the configuration described previously (see section II),
let us discuss some numerical results. To begin with, we explore the impact on
the CLP of covering the dielectric grating with graphene. The ratio of the
pressure with graphene to that without graphene across various values of $\mu$
and $f$, for separation distances ranging from 60 nm to 8 $\mu$m, at
temperatures $T=300$ K and $T=10$ K is depicted in Fig. 2.

At $T=300$ K and for the slab configuration ($f=1$), we observe in Fig. 2(a),
that at short distances, below 60 nm, the CLP with graphene is approximately
10% higher than without graphene for $\mu=0$ and $\mu=0.2$ eV (solid lines
with * and + markers), and 17% higher for $\mu=0.5$ eV (solid line with square
marker). This enhancement peaks at $d=500$ nm before declining as $d$
increases. Beyond this separation, the CLP enhancement is almost identical for
the three values of $\mu$. Figure 2(b) shows the ratio when the temperature is
reduced to 10 K for the same range of $d$. The enhancement remains $<$ 20%.
For $\mu$ = 0.2 eV and 0 eV, no peaks are observed. The ratio decreases to 5%
at $d\approx$ 1$\mu$m.

As the system cools down to $T=10$ K, we observe a behavior similar to that at
$T=300$ K at short separation distances with minor variations. However, as the
separation distance increases, the influence of graphene diminishes and
becomes negligible, with this effect being less than 5% for $\mu=0$ eV and
$\mu=0.2$ eV.

Figure 2: Ratio of the CLP with graphene to CLP without graphene for different
values of $\mu$ and $f$ at $T$ = 300 K (a) and $T$ = 10 K (b). Here, $D=1$
$\mu$m, $h_{1}=300$ nm, and $h_{2}=100$ nm.

On the other hand, the effect of graphene becomes more pronounced for the
grating configuration. Considering $T=300$ K and $f$=0.2, as shown Fig. 2(a),
we clearly see significant changes on the CLP, with approximately a 100%
increase for $\mu=0$ eV (dotted line with *), 110% for $\mu=0.2$ eV (dotted
line with +), and 130% for $\mu=0.5$ eV (dotted line with square) at short
separation distances (i.e. below $d=60$ nm). Similar trends persist at $T=10$
K (Fig. 2(b)), where the relative change remains considerable, around 85% for
$\mu=0$ eV, 105% for $\mu=0.2$ eV, and 130% for $\mu=0.5$ eV. This enhancement
decreases with increasing separation distance, reaching approximately 40% at 1
$\mu$m for $T=300$ K, irrespective of the chemical potential. However, at
$T=10$ K, the impact of graphene at this separation distance varies with the
chemical potential, with enhancements of 13%, 20%, and 32% observed for
$\mu=0$ eV, $\mu=0.2$ eV, and $\mu=0.5$ eV, respectively. Additionally, the
enhancement of the CLP by graphene continues to diminish with increasing
separation distance at both temperatures.

Figure 3: Casimir-Lifshitz pressure at $T=300$ K (a) and $T=10$ K (b), for
$\mu=0$ eV and different filling fractions $f$ = 0, 0.2, 0.5, 0.8, and 1. The
Lifshitz limit for metals, $P_{\text{Lif}}$, is shown as a black dashed line.

To gain more insight into the order of magnitude of the CLP and how it varies
with the geometric parameters and temperature, we plot the pressure $P$ as a
function of $d$ on a log-log scale for $\mu=0$ eV and at $T=300$ K [Fig. 3(a)]
and at $T=10$ K [Fig. 3(b)] . These plots cover distances from 60 nm to 8
$\mu$m for different filling fractions $f=0,0.2,0.5,0.8$, and 1. Notably, when
$f=0$, the configuration corresponds to the graphene sheet located on top of a
vacuum gap 300 nm from the top surface of the SiO2 layer with thickness
$h_{2}$ = 100 nm, supported by a Silicon half-space. We observe that the CLP
increases with the filling fraction $f$ regardless of temperature. Moreover,
at $T=300$ K, the CLP reaches the Lifshitz limit for metals
($P_{\text{Lif}}=-\frac{k_{B}T\zeta(3)}{8\pi d^{3}}$) at nearly 6 $\mu$m (for
all filling fractions), as illustrated clearly in Fig. 3(a). However, at
$T=10$ K, depicted in Fig. 3(b), this limit remains unattained within the
explored separation distances due to the larger thermal wavelength,
$\lambda_{T}=\frac{\hbar c}{k_{B}T}$, which is approximately 229 $\mu$m.

###  IV.2 Chemical potential and geometry effects

Next, we investigate the impact of the chemical potential combined with the
geometry of the grating. For that, we analyze the ratio of the CLP at
$\mu=0.2$ eV and $0.5$ eV to that at $\mu=0$ eV, for various filling fractions
$f=0,0.2,0.5,0.8$, and 1, and spanning distances from 60 nm to 8 $\mu$m.

At $T=300$ K and $\mu=0.2$ eV, Fig. 4(a) shows that having a non-zero chemical
potential always increases the pressure for any value of $f$. This
augmentation is larger for smaller filling fraction. The ratio peaks at around
80 nm and decreases with distance. This effect is most pronounced when $f=0$,
reaching a maximum increase of $\approx$9% and for $f=1$ (representing a slab
configuration) the maximum increase is reduced to only 1%.

As the chemical potential increases to $\mu=0.5$ eV, as shown in Fig. 4(b),
the effect becomes more significant, with an increase of 32% for $f=0$, 16%
for $f=0.2$ at $d\approx$ 100 nm. The maximum ratio also becomes larger for
other values of $f$.

Figure 4: The ratio of CLP (calculated using Eq. (1)) at $\mu=0.2$ eV (a) and
$\mu=0.5$ eV (b) to that at $\mu=0$ eV, at $T=300$ K, for various filling
fractions $f$ = 0, 0.2, 0.5, 0.8, and 1.

Figure 5 shows the results when the temperature is reduced to 10 K. The most
notable change is that the effect of mu becomes stronger. The ratio
$P(\mu)/P(\mu=0)$ becomes larger for the chosen parameters. In particular, the
maximum ratio increases for all values of f compared to 300 K. For example,
the peak ratio for $f$ = 0 is 1.42 and 1.85 for $\mu$ = 0.2 eV [Fig. 5(a)] and
0.5 eV [Fig. 5(b)] respectively at 10 K. Furthermore, it is apparent that the
distance at which the ratio attains peak value is shifted towards larger
values compared to 300K.

Figure 5: The ratio of CLP (calculated using Eq. (1)) at $\mu=0.2$ eV (a) and
$\mu=0.5$ eV (b) to that at $\mu=0$ eV, at $T=10$ K, for various filling
fractions $f$ = 0, 0.2, 0.5, 0.8, and 1.

Figure 6: Normalized CLP for different filling fractions $f$ = 0, 0.2, 0.5,
0.8, and 1 (if $\sigma_{g}\neq 0$) relative to the CLP for $f=1$ with graphene
(a) and without graphene (b) at $T=300$ K and for $\mu=0$ eV.

To further study the impact of geometry, we analyze the ratio of the CLP for a
graphene-covered dielectric grating to that of both a graphene-covered slab
and a bare slab (without graphene). In Fig. 6(a), where $T=300$ K and $\mu=0$
eV, the CLP between the graphene-covered grating and the gold half-space
remains lower than that between a graphene-covered slab and a gold half-space
when the separation distance is less than 4 $\mu$m. Beyond 4 $\mu$m, the
effect of $f$ diminishes, with the ratio approaching 1, since the two
structures have the same asymthotique Lifshitz limit. Similarly, at $T$ = 10
K, as depicted in Fig. 7(a), we observe a comparable trend with a greater
saturation distance.

On the other hand, the CLP for the graphene-coated dielectric grating ($f\neq$
1) is initially lower than that for the bare slab at small separation
distances, as illustrated in Fig. 6(b). However, as the separation distance
$d$ increases, this trend reverses. For $f=0.8$, the CLP exceeds that of the
bare slab beyond 300 nm, while for $f=0.5$, this occurs beyond 1 $\mu$m. As
$f$ increases, the cross-over distance shifts to larger values. The crossover
no longer occurs for $T=10$ K, as it is clear from Fig. 7(b) where the CLP for
the graphene-coated grating ($f\neq$ 1) is lower that that of the bare slab.
It is noteworthy that when the slab is covered with graphene ($f=1$), the
pressure is always higher compared to that without graphene whether at $T=300$
K or $T=10$ K.

Figure 7: Normalized CLP for different filling fractions $f$ = 0, 0.2, 0.5,
0.8, and 1 (if $\sigma_{g}\neq 0$) relative to the CLP for $f=1$ with graphene
(a) and without graphene (b) at $T=10$ K and for $\mu=0$ eV.

###  IV.3 Thermal effect and Casimir force gradient

Now we explore the thermal impact of graphene by examining the ratio of the
CLP at $T=300$ K to that at $T=10$ K. Initially, for $\mu=0$ eV and $f=0$, a
notable thermal effect emerges around 300 nm, where the ratio of approximately
1.7 (Fig. 8(a)). As we increase the filling fraction, this peak gradually
diminishes in height and shifts towards larger separation distances. For
instance, the local peak decreases to approximately 1.4 for $f=0.2$ (at
$d$=500 nm) and 1.15 at $d=1\mu$m for $f=1$. After this peak (for every $f$),
the ratio decreases, before eventually experiencing a sharp increase for
distances $\approx 6\mu$m. The sharp rise is expected because the pressure at
$T=300$ K reaches the Lifshitz limit (see Fig. 3(a)) while remaining below it
at $T=10$ K (see Fig. 3(b)). In this limiting case, using the expression
$P_{\text{Lif}}=k_{B}T\zeta(3)/(8\pi d^{3})$ yields a ratio of 30.
Consequently, it is expected that the ratio continues to increase until
reaching the thermal wavelength, approximately 229 $\mu$m, beyond which the
value levels off.

Moreover, this thermal effect decreases with the chemical potential, as
demonstrated in Fig. 8(b) and 8(c) for $\mu=0.2$ and 0.5, respectively. This
is due to the fact that the graphene conductivity (6) only depends weakly on
the temperature $T\approx 10^{-3}eV$ when $\mu$ is larger than $T$.

Figure 8: Ratio of the CLP at $T=300$ K to that at $T=10$ K for $\mu=0$ eV
(a), $\mu=0.2$ eV (b), and $\mu=0.5$ eV (c), for different filling fractions
$f$ = 0, 0.2, 0.5, 0.8, and 1.

Finally, we consider the gradient of the Casimir force between a gold sphere
and a dielectric grating covered with graphene. According to the Proximity
Force Approximation (PFA) [14, 11], the Casimir pressure $P$ between two
parallel plates is proportional to the gradient of the Casimir force in the
sphere-plate geometry, and it can be expressed as $F^{\prime}=2\pi RP$, where
$R$ is the sphere radius. The latter configuration is used in experiments to
avoid the difficulty in keeping two planar bodies parallel at short
separations. We considered a gold sphere with a radius of 40 $\mu$m at $T=300$
K and $T=10$ K, and calculated the gradient over distances ranging from 60 nm
to 1 $\mu$ m for two configurations: a slab ($f=1$) and a grating with $f=0.2$
and $\mu=0$ eV.

Our analysis revealed that the effect of the graphene on the force gradient is
significantly more pronounced for a grating compared to solid SiO2,
particularly at short separation distances for both $T$ = 10 K and 300 K. the
force gradient at $d$= 200 nm increases by 94% and 55% respectively for $T$ =
300 K and 10K when graphene is added to an SiO2 grating with $f$ = 0.2.

Figure 9: Gradient of the Casimir force between a gold sphere with radius $R$
= 40 $\mu$m and a graphene-coated fused silica grating with the following
parameters: period $D$ = 1 $\mu$m, depth $h_{1}$ = 300 nm, and slab thickness
$h_{2}$ = 100 nm, at $T=300$ K (a) and $T=10$ K (b). for different values of
$\mu$ and $f$.

##  V Conclusion

In conclusion, we have investigated the Casimir-Lifshitz pressure between a
graphene-coated dielectric grating and a gold half-space using an adapted FMM.
Our results reveal that covering the dielectric grating with graphene leads to
a significant increase in pressure, with enhancements up to 130% for $\mu=0.5$
eV and $f=0.2$, compared to only 17% when covering a slab with graphene, over
a wide range of temperatures from 10K to 300K. Additionally, we have shown
that the CLP depends strongly on the chemical potential of graphene, with
maximal effects observed at lower filling fractions ($f$ = 0.2 in our case)
for both $T$= 300 K and 10 K. Notably, our findings indicate that the pressure
for a graphene-coated dielectric grating can surpass that of a bare slab at
$T=300$ K at certain separation distances even when $f$ is small. Furthermore,
we have identified a thermal effect between 300 nm and 400 nm, which
diminishes with increasing filling factor and chemical potential. Lastly, we
have presented the Casimir force gradient between a gold sphere and a
graphene-coated dielectric grating, a configuration that is used for
experimental measurements.

###### Acknowledgements.

The work described in this paper was supported by a grant ”CAT” from the
ANR/RGC Joint Research Scheme sponsored by the French National Research Agency
(ANR) and the Research Grants Council (RGC) of the Hong Kong Special
Administrative Region, China (Project No. A-HKUST604/20).

##

## APPENDIX: SCATTERING MATRIX CALCULATION FOR GRAPHENE-COVERED GRATING USING
THE FOURIER MODAL METHOD

In this appendix, we will describe the calculation of the scattering matrix
using the Fourier Modal Method for dielectric gratings coated with graphene,
employing a zero-thickness model. This approach simplifies the representation
of graphene and facilitates its direct inclusion into the boundary conditions
of the FMM. The structure under study is depicted in Fig. 10, where a graphene
sheet is positioned at $z=0$. Four distinct zones can be distinguished : (i)
zone 1 ($z<0$) is characterized by $\varepsilon_{1}$, (ii) zone 2
($0<z<h_{1}$) contains a lamellar grating with a period $D$ along the
$x$-axis, a relative dielectric permittivity defined denoted
$\varepsilon_{2}(x)$ and a thickness $h_{1}$, (iii) zone 3
$(h_{1}<z<h_{1}+h_{2})$, is a homogeneous slab with permittivity
$\varepsilon_{3}$ and (iv) zone 4 ($z>h_{1}+h_{2}$) is a semi-infinite medium
characterized by $\varepsilon_{4}$.

Figure 10: Geometry of the FMM calculation: A dielectric grating characterized
by dielectric permittivity $\varepsilon_{2}(x)$, with period $D$, width $a$,
and depth $h_{1}$, covered with a graphene sheet and placed on a top of a slab
with dielectric permittivity $\varepsilon_{3}$ and thickness $h_{2}$, which in
turn rests on a dielectric half-space with dielectric permittivity
$\varepsilon_{4}$.

###  V.1 Electromagnetic Fields

Due to periodicity along the $x-$axis, the electromagnetic fields in each
homogeneous zone ($i=1,3,4$) can be expanded into the so-called Rayleigh
expansion as follows:

$\left\\{\begin{aligned} {\bf E}^{(i)}({\bf
R},\omega)&=\sum_{p,\phi}\int_{-\frac{\pi}{D}}^{\frac{\pi}{D}}\dfrac{\mathrm{d}k_{x}}{2\pi}\sum_{n\in\mathbb{Z}}\int_{-\infty}^{+\infty}\dfrac{\mathrm{d}k_{y}}{2\pi}\\\
&\times e^{i{\bf K}_{n}^{(i)\phi}\cdot\bf{R}}\hat{{\bf e}}_{p}^{(i)\phi}({\bf
k}_{n},\omega)E_{p}^{(i)\phi}({\bf k}_{n},\omega),\\\ {\bf B}^{(i)}({\bf
R},\omega)&=\frac{\sqrt{\varepsilon_{i}(\omega)}}{c}\sum_{p,\phi}\int_{-\frac{\pi}{D}}^{\frac{\pi}{D}}\dfrac{\mathrm{d}k_{x}}{2\pi}\sum_{n\in\mathbb{Z}}\int_{-\infty}^{+\infty}\dfrac{\mathrm{d}k_{y}}{2\pi}\\\
&\times e^{i{\bf K}_{n}^{(i)\phi}\cdot\bf{R}}(-1)^{p}{\hat{{\bf
e}}_{S(p)}^{(i){\bf\phi}}}({\bf k}_{n},\omega)E_{p}^{(i)\phi}({\bf
k}_{n},\omega),\end{aligned}\right.$ (21)

where $p$ denotes the index of polarization ($p$=1,2 for TE and TM
respectively), ${\bf R}=({\bf r},z)$, $S(p)$ is a function ($S(1)=2$ and
$S(2)=1$), and

$\left\\{\begin{aligned} &\hat{\bf{e}}_{\rm
TE}^{(i)\phi}({\bf{k}}_{n},\omega)=\frac{1}{k_{n}}(-k_{y}\hat{\bf{e}}_{x}+k_{xn}\hat{\bf{e}}_{y}),\\\
&\hat{\bf{e}}_{\rm
TM}^{(i)\phi}({\bf{k}}_{n},\omega)=\frac{c}{\omega\sqrt{\varepsilon_{i}(\omega)}}(-k_{n}\hat{\bf{e}}_{z}+\phi
k_{zn}^{i}\hat{\bf{k}}_{n}).\end{aligned}\right.$ (22)

Here, $\hat{\bf{e}}_{x}$, $\hat{\bf{e}}_{y}$ and $\hat{\bf{e}}_{z}$ are unit
vectors in the $(x,y,z)$ Cartesian basis,
$\hat{\bf{k}}_{n}={\bf{k}}_{n}/k_{n}$, $\phi$ is the direction of propagation
of the waves $(+,-)$ along the $z$-axis for the incident and the reflected
fields, respectively, and ${\bf K}_{n}^{(i)\phi}=({\bf k}_{n},\phi
k_{zn}^{(i)})$.

For the different amplitudes in these zones, we use the notations :

$\left\\{\begin{aligned} E^{(1)+}&=I,\;\;\;\;E^{(1)-}=R,\\\
E^{(3)+}&=C,\;\;\;\;E^{(3)-}=C^{\prime},\\\ E^{(4)+}&=T,\;\;\;\;E^{(4)-}=J,\\\
\end{aligned}\right.$ (23)

In the periodic region (zone 2), the electric field assumes the following
form:

${\bf E}^{(2)}({\bf
R},\omega)=\int_{-\frac{\pi}{D}}^{\frac{\pi}{D}}\dfrac{\mathrm{d}k_{x}}{2\pi}\sum_{n\in\mathbb{Z}}\int_{-\infty}^{+\infty}\dfrac{\mathrm{d}k_{y}}{2\pi}e^{i{\bf
k}_{n}\cdot\bf{r}}{\bf E}^{(2)}(z,{\bf k}_{n},\omega).$ (24)

In the actual computation, we will truncate the sum ranging from $-\infty$ to
$\infty$, retaining only $2N+1$ Fourier harmonics, $N$ is termed the
truncation order. In order to obtain the amplitudes ${\bf E}^{(2)}(z,{\bf
k}_{n},\omega)$, we have to solve Maxwell’s equations within this zone:

$\left\\{\begin{aligned}
\partial_{y}E_{z}-\partial_{z}E_{y}=ik_{0}\tilde{H}_{x}\\\
\partial_{z}E_{x}-\partial_{x}E_{z}=ik_{0}\tilde{H}_{y}\\\
\partial_{x}E_{y}-\partial_{y}E_{x}=ik_{0}\tilde{H}_{z}\\\
\end{aligned}\right.\left\\{\begin{aligned}
\partial_{y}\tilde{H}_{z}-\partial_{z}\tilde{H}_{y}=-i\varepsilon
k_{0}E_{x}\\\
\partial_{z}\tilde{H}_{x}-\partial_{x}\tilde{H}_{z}=-i\varepsilon
k_{0}E_{y}\\\
\partial_{x}\tilde{H}_{y}-\partial_{y}\tilde{H}_{x}=-i\varepsilon
k_{0}E_{z},\\\ \end{aligned}\right.$ (25)

with $k_{0}=\omega/c$, and $\tilde{H}=Z_{0}H$.

Eliminating the $z$ components from Eq. (25), we obtain two systems of
equations involving only the parallel components over which the boundary
conditions hold:

$\displaystyle\partial_{z}\begin{pmatrix}E_{x}\\\
E_{y}\end{pmatrix}=\begin{pmatrix}-\dfrac{i}{k_{0}}\partial_{x}\dfrac{1}{\varepsilon(x)}\partial_{y}&ik_{0}+\dfrac{i}{k_{0}}\partial_{x}\dfrac{1}{\varepsilon(x)}\partial_{x}\\\
-ik_{0}-\dfrac{i}{k_{0}}\partial_{y}\dfrac{1}{\varepsilon(x)}\partial_{y}&\dfrac{i}{k_{0}}\partial_{y}\dfrac{1}{\varepsilon(x)}\partial_{x}\\\
\end{pmatrix}$ (26) $\displaystyle\times\begin{pmatrix}\tilde{H}_{x}\\\
\tilde{H}_{y}\end{pmatrix}$

$\displaystyle\partial_{z}\begin{pmatrix}\tilde{H}_{x}\\\
\tilde{H}_{y}\end{pmatrix}=\begin{pmatrix}\dfrac{i}{k_{0}}\partial_{x}\partial_{y}&-ik_{0}\varepsilon(x)-\dfrac{i}{k_{0}}\partial_{x}\partial_{x}\\\
ik_{0}\varepsilon(x)+\dfrac{i}{k_{0}}\partial_{y}\partial_{y}&-\dfrac{i}{k_{0}}\partial_{x}\partial_{y}\\\
\end{pmatrix}$ (27) $\displaystyle\times\begin{pmatrix}E_{x}\\\
E_{y}\end{pmatrix}$

Subsequently, following the method elaborated in more detail, for instance, in
[41], we transform these equations into Fourier space and get :

$\left\\{\begin{aligned}
\partial_{z}\bm{\mathcal{E}}=\mathbb{F}\bm{\mathcal{\tilde{H}}}\\\
\partial_{z}\bm{\mathcal{\tilde{H}}}=\mathbb{G}\bm{\mathcal{E}}\end{aligned}\right.$
(28)

where $\bm{\mathcal{E}}=[\bm{\mathcal{E}}_{x},\bm{\mathcal{E}}_{y}]^{t}$ and
$\bm{\mathcal{\tilde{H}}}=[\bm{\mathcal{\tilde{H}}}_{x},\bm{\mathcal{\tilde{H}}}_{y}]^{t}$
are vectors containing the Fourier components of the electric and magnetic
parallel components of the field in the grating zone. Moreover,

$\mathbb{F}=\begin{pmatrix}\dfrac{i\beta}{k_{0}}\alpha\left\llbracket\varepsilon\right\rrbracket^{-1}&ik_{0}\mathbbm{1}-\dfrac{i\alpha}{k_{0}}\left\llbracket\varepsilon\right\rrbracket^{-1}\alpha\\\
-ik_{0}\mathbbm{1}+\dfrac{i\beta^{2}}{k_{0}}\left\llbracket\varepsilon\right\rrbracket^{-1}&-\dfrac{i\beta}{k_{0}}\left\llbracket\varepsilon\right\rrbracket^{-1}\alpha\\\
\end{pmatrix},$ (29)

and

$\mathbb{G}=\begin{pmatrix}-\dfrac{i\beta}{k_{0}}\alpha&-ik_{0}\left\llbracket\varepsilon\right\rrbracket+\dfrac{i\alpha^{2}}{k_{0}}\\\
ik_{0}\left\llbracket\varepsilon^{-1}\right\rrbracket^{-1}-\dfrac{i\beta^{2}}{k_{0}}&\dfrac{i\beta}{k_{0}}\alpha\\\
\end{pmatrix}.$ (30)

Here, $\left\llbracket\varepsilon\right\rrbracket$ is the Toeplitz matrix such
that $\left\llbracket\varepsilon\right\rrbracket_{mn}=\varepsilon_{m-n}$,
$\beta$ is a scalar, and $\alpha={\rm diag}(k_{xn})_{n}$.

Using the coupled equations (28), we deduce the equation for
$\bm{\mathcal{E}}$ alone :

$\partial^{2}_{z}\bm{\mathcal{E}}=\mathbb{F}\mathbb{G}\bm{\mathcal{E}}=\mathbb{P}\mathbb{D}^{2}\mathbb{P}^{-1}\bm{\mathcal{E}},$
(31)

where $\mathbb{P}$ and $\mathbb{D}^{2}$ contain, respectively, the
eigenvectors and eigenvalues of the matrix $\mathbb{FG}$, each having
dimensions of $2(2N+1)\times 2(2N+1)$ and take the following form:

$\mathbb{P}=\begin{pmatrix}\mathbb{P}^{(11)}&\mathbb{P}^{(12)}\\\
\mathbb{P}^{(21)}&\mathbb{P}^{(22)}\\\
\end{pmatrix},\mathbb{D}=\begin{pmatrix}\mathbb{D}^{(11)}&\mathbb{0}\\\
\mathbb{0}&\mathbb{D}^{(22)}\\\ \end{pmatrix}.$ (32)

The solution of the set of equations (28) can then be written :

$\left\\{\begin{aligned}
\bm{\mathcal{E}}&=\mathbb{P}(e^{\mathbb{D}z}\bm{\mathcal{A}}+e^{\mathbb{-D}z}\bm{\mathcal{B}})\\\
\bm{\mathcal{\tilde{H}}}&=\mathbb{P^{\prime}}(e^{\mathbb{D}z}\bm{\mathcal{A}}-e^{\mathbb{-D}z}\bm{\mathcal{B}}),\end{aligned}\right.$
(33)

where $\bm{\mathcal{A}}=[A_{xn},A_{yn}]^{t}$ and
$\bm{\mathcal{B}}=[B_{xn},B_{yn}]^{t}$ ($n\in[-N,N]$) are arbitrary constant
vectors, and $\mathbb{P^{\prime}}=\mathbb{F}^{-1}\mathbb{P}\mathbb{D}$.

###  V.2 Boundary conditions

After expressing the fields in all zones, we are going to enforce the boundary
conditions at their interfaces. For both analytical and numerical convenience,
we introduce an additional phase factor in the expressions of the fields
within zones 3 and 4. Specifically in zone 3, we take the $z$ dependance of
the fields under the form $e^{ik^{(i)\phi}(z-h_{1})}$, while in zone 4, we we
take it under the form $e^{ik^{(i)\phi}(z-h_{1}-h_{2})}$. These adjustments
facilitate the calculations and can be easily reverted at the conclusion of
the analysis. For this reason, we will first obtain the scattering matrix with
this simplified phase $\mathbb{\tilde{S}}$, and then multiply it by the
accumulated phase to obtain the actual scattering matrix $\mathbb{S}$ (see Eq.
(8)).

Using the zero thickness model for the graphene sheet and applying the
boundary conditions for the electric field components at the interface ($z$=0)
between zones 1 and 2, in the cartesian coordinates system, we obtain (after
projection on the Fourier basis):

$\left(\begin{array}[]{c}-\frac{k_{y}}{k_{n}}\left(I_{1n}+R_{1n}\right)+\frac{c}{\sqrt{\varepsilon_{1}}\omega}k_{zn}^{(1)}\frac{k_{xn}}{k_{n}}\left(I_{2n}-R_{2n}\right)\\\
\frac{k_{xn}}{k_{n}}\left(I_{1n}+R_{1n}\right)+\frac{c}{\sqrt{\varepsilon_{1}}\omega}k_{zn}^{(1)}\frac{k_{y}}{k_{n}}\left(I_{2n}-R_{2n}\right)\end{array}\right)=\left(\begin{array}[]{l}\mathbb{P}_{nm}^{(11)}\left(A_{xm}+B_{xm}\right)+\mathbb{P}_{nm}^{(12)}\left(A_{ym}+B_{ym}\right)\\\
\mathbb{P}_{nm}^{(21)}\left(A_{xm}+B_{xm}\right)+\mathbb{P}_{nm}^{(22)}\left(A_{ym}+B_{ym}\right)\end{array}\right).$
(34)

Doing the same for the magnetic field components, we have:

$\displaystyle\left(\begin{array}[]{l}\mathbb{P}_{nm}^{\prime(11)}\left(A_{xm}-B_{xm}\right)+\mathbb{P^{\prime}}_{nm}^{(12)}\left(A_{ym}-B_{ym}\right)+\frac{c}{\omega}k_{zn}^{(1)}\frac{k_{xn}}{k_{n}}\left(I_{1n}-R_{1n}\right)+\sqrt{\varepsilon_{1}}\frac{k_{y}}{k_{n}}\left(I_{2n}+R_{2n}\right)\\\
\mathbb{P}_{nm}^{\prime(21)}\left(A_{xm}-B_{xm}\right)+\mathbb{P}_{nm}^{\prime(22)}\left(A_{ym}-B_{ym}\right)+\frac{c}{\omega}k_{zn}^{(1)}\frac{k_{y}}{k_{n}}\left(I_{1n}-R_{1n}\right)-\sqrt{\varepsilon_{1}}\frac{k_{xn}}{k_{n}}\left(I_{2n}+R_{2n}\right)\end{array}\right)$
(35) $\displaystyle=\sigma_{\rm
g}Z_{0}\left(\begin{array}[]{c}\frac{k_{xn}}{k_{n}}\left(I_{1n}+R_{1n}\right)+\frac{c}{\sqrt{\varepsilon_{1}}\omega}k_{zn}^{(1)}\frac{k_{y}}{k_{n}}\left(I_{2n}-R_{2n}\right)\\\
\frac{k_{y}}{k_{n}}\left(I_{1n}+R_{1n}\right)-\frac{c}{\sqrt{\varepsilon_{1}}\omega}k_{zn}^{(1)}\frac{k_{xn}}{k_{n}}\left(I_{2n}-R_{2n}\right)\end{array}\right).$

At the interface $z=h_{1}$ we have the following equations for the electric
field

$\displaystyle\left(\begin{array}[]{c}-\frac{k_{y}}{k_{n}}\left(C_{1n}+C_{1n}^{\prime}\right)+\frac{c}{\sqrt{\varepsilon_{3}}\omega}k_{zn}^{(3)}\frac{k_{xn}}{k_{n}}\left(C_{2n}-C_{2n}^{\prime}\right)\\\
\frac{k_{xn}}{k_{n}}\left(C_{1n}+C_{1n}^{\prime}\right)+\frac{c}{\sqrt{\varepsilon_{3}}\omega}k_{zn}^{(3)}\frac{k_{y}}{k_{n}}\left(C_{2n}-C_{2n}^{\prime}\right)\end{array}\right)$
(36)
$\displaystyle=\left(\begin{array}[]{l}\mathbb{P}_{nm}^{(11)}\left(e^{\mathbb{D}_{mm}^{(11)}h_{1}}A_{xm}+e^{-\mathbb{D}_{mm}^{(11)}h_{1}}B_{xm}\right)+\mathbb{P}_{nm}^{(12)}\left(e^{\mathbb{D}_{mm}^{(22)}h_{1}}A_{ym}+e^{-\mathbb{D}_{m}^{(22)}h_{1}}B_{ym}\right)\\\
\mathbb{P}_{nm}^{(21)}\left(e^{\mathbb{D}_{mm}^{(11)}h_{1}}A_{x,m}+e^{-\mathbb{D}_{mm}^{(11)}h_{1}}B_{xm}\right)+\mathbb{P}_{nm}^{(22)}\left(e^{\mathbb{D}_{mm}^{(22)}h_{1}}A_{ym}+e^{-\mathbb{D}_{mm}^{(22)}h_{1}}B_{ym}\right)\end{array}\right),$

and the boundary conditions for the magnetic field are as follows:

$\displaystyle\left(\begin{array}[]{l}-\frac{c}{\omega}k_{zn}^{(3)}\frac{k_{xn}}{k_{n}}\left(C_{1n}-C_{1n}^{\prime}\right)-\sqrt{\varepsilon_{3}}\frac{k_{y}}{k_{n}}\left(C_{2n}+C_{2n}^{\prime}\right)\\\
-\frac{c}{\omega}k_{zn}^{(3)}\frac{k_{y}}{k_{n}}\left(C_{1n}-C_{1n}^{\prime}\right)+\sqrt{\varepsilon_{3}}\frac{k_{xn}}{k_{n}}\left(C_{2n}+C_{2n}^{\prime}\right)\end{array}\right)$
(37)
$\displaystyle\quad=\left(\begin{array}[]{l}\mathbb{P}_{nm}^{\prime(11)}\left(e^{\mathbb{D}_{mm}^{(11)}h_{1}}A_{xm}-e^{-\mathbb{D}_{mm}^{(11)}h_{1}}B_{xm}\right)+\mathbb{P^{\prime}}_{nm}^{(12)}\left(e^{\mathbb{D}_{mm}^{(22)}h_{1}}A_{ym}-e^{-\mathbb{D}_{mm}^{(22)}h_{1}}B_{ym}\right)\\\
\mathbb{P}_{nm}^{\prime(21)}\left(e^{\mathbb{D}_{mm}^{(11)}h_{1}}A_{xm}-e^{-\mathbb{D}_{mm}^{(11)}h_{1}}B_{xm}\right)+\mathbb{P}_{nm}^{\prime(22)}\left(e^{\mathbb{D}_{mm}^{(22)}h_{1}}A_{ym}-e^{-\mathbb{D}_{mm}^{(22)}h_{1}}B_{ym}\right)\end{array}\right).$

Likewise, at the final interface, $z=h_{1}+h_{2}$, we obtain

$\displaystyle\left(\begin{array}[]{c}-\frac{k_{y}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{1n}+e^{-ik_{zn}^{(3)}h_{2}}C_{1n}^{\prime}\right)+\frac{c}{\sqrt{\varepsilon_{3}}\omega}k_{zn}^{(3)}\frac{k_{xn}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{2n}-e^{-ik_{zn}^{(3)}h_{2}}C_{2n}^{\prime}\right)\\\
\frac{k_{xn}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{1n}+e^{-ik_{zn}^{(3)}h_{2}}C_{1n}^{\prime}\right)+\frac{c}{\sqrt{\varepsilon_{3}}\omega}k_{zn}^{(3)}\frac{k_{y}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{2n}-e^{-ik_{zn}^{(3)}h_{2}}C_{2n}^{\prime}\right)\end{array}\right)$
(38)
$\displaystyle=\left(\begin{array}[]{c}-\frac{k_{y}}{k_{n}}\left(T_{1n}+J_{1n}\right)+\frac{c}{\sqrt{\varepsilon_{4}}\omega}k_{zn}^{(4)}\frac{k_{xn}}{k_{n}}\left(T_{2n}-J_{2n}\right)\\\
\frac{k_{xn}}{k_{n}}\left(T_{1n}+J_{1n}\right)+\frac{c}{\sqrt{\varepsilon_{4}}\omega}k_{zn}^{(4)}\frac{k_{y}}{k_{n}}\left(T_{2n}-J_{2n}\right)\end{array}\right)$

and

$\displaystyle\left(\begin{array}[]{l}-\frac{c}{\omega}k_{zn}^{(3)}\frac{k_{xn}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{1n}-e^{-ik_{zn}^{(3)}h_{2}}C_{1n}^{\prime}\right)-\sqrt{\varepsilon_{3}}\frac{k_{y}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{2n}+e^{-ik_{zn}^{(3)}h_{2}}C_{2n}^{\prime}\right)\\\
-\frac{c}{\omega}k_{zn}^{(3)}\frac{k_{y}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{1n}-e^{-ik_{2n}^{(3)}h_{2}}C_{1n}^{\prime}\right)+\sqrt{\varepsilon_{3}}\frac{k_{xn}}{k_{n}}\left(e^{ik_{zn}^{(3)}h_{2}}C_{2n}+e^{-ik_{zn}^{(3)}h_{2}}C_{2n}^{\prime}\right)\end{array}\right)$
(39)
$\displaystyle=\left(\begin{array}[]{l}-\frac{c}{\omega}k_{zn}^{(4)}\frac{k_{xn}}{k_{n}}\left(T_{1n}-J_{1n}\right)-\sqrt{\varepsilon_{4}}\frac{k_{y}}{k_{n}}\left(T_{2n}+J_{2n}\right)\\\
-\frac{c}{\omega}k_{zn}^{(4)}\frac{k_{y}}{k_{n}}\left(T_{1n}-J_{1n}\right)+\sqrt{\varepsilon_{4}}\frac{k_{xn}}{k_{n}}\left(T_{2n}+J_{2n}\right)\end{array}\right).$

###  V.3 Scattering matrix

Finally, we can now write the previous equations in matrix compact form as

$\displaystyle\left(\begin{array}[]{l}\bm{\mathcal{R}}\\\
\bm{\mathcal{A}}\end{array}\right)$
$\displaystyle=\mathbb{\tilde{S}}_{1}\left(\begin{array}[]{l}\bm{\mathcal{I}}\\\
\bm{\mathcal{B}}\end{array}\right),\quad\left(\begin{array}[]{l}\bm{\mathcal{B}}\\\
\bm{\mathcal{C}}\end{array}\right)=\mathbb{\tilde{S}}_{2}\left(\begin{array}[]{l}\bm{\mathcal{A}}\\\
\bm{\mathcal{C}}^{\prime}\end{array}\right),$ (40) $\displaystyle{\rm and}$
$\displaystyle\quad\left(\begin{array}[]{l}\bm{\mathcal{C}^{\prime}}\\\
\bm{\mathcal{T}}\end{array}\right)=\mathbb{\tilde{S}}_{3}\left(\begin{array}[]{l}\bm{\mathcal{C}}\\\
\bm{\mathcal{J}}\end{array}\right),$

where the explicit expressions of $\mathbb{\tilde{S}}_{1}$,
$\mathbb{\tilde{S}}_{2}$, and $\mathbb{\tilde{S}}_{3}$ are already given in
section III.

Furthermore, to derive the actual scattering matrix, we incorporate a phase
correction, as previously mentioned. This adjustment is implemented in Eq.
(8).

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