# Hochschild homology, trace map and $\zeta$-cycles

Alain Connes  Collège de France and IHES, France<EMAIL_ADDRESS>and
Caterina Consani  Department of Mathematics, Johns Hopkins University,
Baltimore MD 21218, USA<EMAIL_ADDRESS>
(Date: and, in revised form, .)

###### Abstract.

In this paper we consider two spectral realizations of the zeros of the
Riemann zeta function. The first one involves all non-trivial (i.e. non-real)
zeros and is expressed in terms of a Laplacian intimately related to the
prolate wave operator. The second spectral realization affects only the
critical zeros and it is cast in terms of sheaf cohomology. The novelty is
that the base space is the Scaling Site playing the role of the parameter
space for the $\zeta$-cycles and encoding their stability by coverings.

###### Key words and phrases:

Riemann zeta function, spectral realization, zeta cycle, Hochschild homology,
adèle class space, Scaling Site

###### 2020 Mathematics Subject Classification:

Primary 11M55, 11M06, 16E40; Secondary 11F72, 58B34

The second author is partially supported by the Simons Foundation
collaboration grant n. 691493.

##  1\. Introduction

In this paper we give a Hochschild homological interpretation of the zeros of
the Riemann zeta function. The root of this result is in the recognition that
the map

$({\mathcal{E}}f)(u)=u^{1/2}\sum_{n>0}f(nu)$

which is defined on a suitable subspace of the linear space of complex-valued
even Schwartz functions on the real line, is a trace in Hochschild homology,
if one brings in the construction the projection
$\pi:{\mathbb{A}}_{\mathbb{Q}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$
from the rational adèles to the adèle classes (see Section 3). In this paper,
we shall consider two spectral realizations of the zeros of the Riemann zeta
function. The first one involves all non-trivial (i.e. non-real) zeros and is
expressed in terms of a Laplacian intimately related to the prolate wave
operator (see Section 4). The second spectral realization is sharper inasmuch
as it affects only the critical zeros. The main players are here the
$\zeta$-cycles introduced in [7], and the Scaling Site [6] as their parameter
space, which encodes their stability by coverings. The $\zeta$-cycles give the
theoretical geometric explanation for the striking coincidence between the low
lying spectrum of a perturbed spectral triple therein introduced (see [7]),
and the low lying (critical) zeros of the Riemann zeta function. The
definition of a $\zeta$-cycle derives, as a by-product, from scale-invariant
Riemann sums for complex-valued functions on the real half-line $[0,\infty)$
with vanishing integral. For any $\mu\in{\mathbb{R}}_{>1}$, one implements the
linear (composite) map

$\Sigma_{\mu}{\mathcal{E}}:{{\mathcal{S}}^{\rm ev}_{0}}\to L^{2}(C_{\mu})$

from the Schwartz space ${{\mathcal{S}}^{\rm ev}_{0}}$ of real valued even
functions $f$ on the real line, with $f(0)=0$, and vanishing integral, to the
Hilbert space $L^{2}(C_{\mu})$ of square integrable functions on the circle
$C_{\mu}={\mathbb{R}}_{+}^{*}/\mu^{\mathbb{Z}}$ of length $L=\log\mu$, where

$(\Sigma_{\mu}g)(u):=\sum_{k\in{\mathbb{Z}}}g(\mu^{k}u).$

The map $\Sigma_{\mu}$ commutes with the scaling action
${\mathbb{R}}^{*}_{+}\ni\lambda\mapsto f(\lambda^{-1}x)$ on functions, while
${\mathcal{E}}$ is invariant under a normalized scaling action on
${{\mathcal{S}}^{\rm ev}_{0}}$. In this set-up one has

######  Definition.

A $\zeta$-cycle is a circle $C$ of length $L=\log\mu$ whose Hilbert space
$L^{2}(C)$ contains $\Sigma_{\mu}{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}})$
as a non dense subspace.

Next result is known (see [7] Theorem 6.4)

######  Theorem 1.1.

The following facts hold

  1. (i)

The spectrum of the scaling action of ${\mathbb{R}}_{+}^{*}$ on the orthogonal
space to $\Sigma_{\mu}{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}})$ in
$L^{2}(C_{\mu})$ is contained in the set of the imaginary parts of the zeros
of the Riemann zeta function $\zeta(z)$ on the critical line
$\Re(z)=\frac{1}{2}$.

  2. (ii)

Let $s>0$ be a real number such that $\zeta(\frac{1}{2}+is)=0$. Then any
circle $C$ whose length is an integral multiple of $\frac{2\pi}{s}$ is a
$\zeta$-cycle, and the spectrum of the action of ${\mathbb{R}}_{+}^{*}$ on
$(\Sigma_{\mu}{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}}))^{\perp}$ contains
$s$.

Theorem 1.1 states that for a countable and dense set of values of
$L\in{\mathbb{R}}_{>0}$, the Hilbert spaces
${\mathcal{H}}(L):=(\Sigma_{\mu}{\mathcal{E}}({{\mathcal{S}}^{\rm
ev}_{0}}))^{\perp}$ are non-trivial and, more importantly, that as $L$ varies
in that set, the spectrum of the scaling action of ${\mathbb{R}}_{+}^{*}$ on
the family of the ${\mathcal{H}}(L)$’s is the set $Z$ of imaginary parts of
critical zeros of the Riemann zeta function. In fact, in view of the proven
stability of $\zeta$-cycles under coverings, the same element of $Z$ occurs
infinitely many times in the family of the ${\mathcal{H}}(L)$’s. This
stability under coverings displays the Scaling Site
${\mathscr{S}}={[0,\infty)\rtimes{{\mathbb{N}}^{\times}}}$ as the natural
parameter space for the $\zeta$-cycles. In this paper, we show (see Section 5)
that after organizing the family ${\mathcal{H}}(L)$ as a sheaf over
${\mathscr{S}}$ and using sheaf cohomology, one obtains a spectral realization
of critical zeros of the Riemann zeta function. The key operation in the
construction of the relevant arithmetic sheaf is given by the action of the
multiplicative monoid ${\mathbb{N}}^{\times}$ on the sheaf of smooth sections
of the bundle $L^{2}$ determined by the family of Hilbert spaces
$L^{2}(C_{\mu})$, $\mu={\exp L}$, as $L$ varies in $(0,\infty)$. For each
$n\in{\mathbb{N}}^{\times}$ there is a canonical covering map $C_{\mu^{n}}\to
C_{\mu}$, where the action of $n$ corresponds to the operation of sum on the
preimage of a point in $C_{\mu}$ under the covering. This action turns the
(sub)sheaf of smooth sections vanishing at $L=0$ into a sheaf
${\mathcal{L}}^{2}$ over ${\mathscr{S}}$. The family of subspaces
$\Sigma_{\mu}{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}})\subset
L^{2}(C_{\mu})$ generates a closed subsheaf
$\overline{\Sigma{\mathcal{E}}}\subset{\mathcal{L}}^{2}$ and one then
considers the cohomology of the related quotient sheaf
${\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}$. In view of the property of
${\mathbb{R}}_{+}^{*}$-equivariance under scaling, this construction
determines a spectral realization of critical zeros of the Riemann zeta
function, also taking care of eventual multiplicities. Our main result is the
following

######  Theorem 1.2.

The cohomology
$H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}})$
endowed with the induced canonical action of ${\mathbb{R}}_{+}^{*}$ is
isomorphic to the spectral realization of critical zeros of the Riemann zeta
function, given by the action of ${\mathbb{R}}_{+}^{*}$, via multiplication
with $\lambda^{is}$, on the quotient of the Schwartz space
${\mathcal{S}}({\mathbb{R}})$ by the closure of the ideal generated by
multiples of $\zeta\left(\frac{1}{2}+is\right)$.

This paper is organized as follows. Section 2 recalls the main role played by
the (image of the) map ${\mathcal{E}}$ in the study of the spectral
realization of the critical zeros of the Riemann zeta function. In Section 3
we show the identification of the Hochschild homology HH0 of the
noncommutative space
${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$ with the
coinvariants for the action of ${\mathbb{Q}}^{\times}$ on the Schwartz
algebra, using the (so-called) “wrong way” functoriality map $\pi_{!}$
associated to the projection
$\pi:{\mathbb{A}}_{\mathbb{Q}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$.
We also stress the relevant fact that the Fourier transform on adèles becomes
canonical after passing to $\text{HH}_{0}$ of the adèle class space of the
rationals. The key Proposition 3.3 describes the invariant part of such
$\text{HH}_{0}$ as the space of even Schwartz functions on the real line and
identifies the trace map with the map ${\mathcal{E}}$. Section 4 takes care of
the two vanishing conditions implemented in the definition of ${\mathcal{E}}$
and introduces the operator $\Delta=H(1+H)$ ($H$ being the generator of the
scaling action of ${\mathbb{R}}^{*}_{+}$ on
${\mathcal{S}}({\mathbb{R}})^{\text{ev}}$) playing the role of the Laplacian
and intimately related to the prolate operator. Finally, Section 5 is the main
technical section of this paper since it contains the proof of Theorem 1.2.

##  2\. The map ${\mathcal{E}}$ and the zeros of the zeta function

The adèle class space of the rationals
${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$ is the natural
geometric framework to understand the Riemann-Weil explicit formulas for
$L$-functions as a trace formula [3]. The essence of this result lies mainly
in the delicate computation of the principal values involved in the
distributions appearing in the geometric (right-hand) side of the semi-local
trace formula of op.cit. (see Theorem 4 for the notations)

$\text{Trace}(R_{\Lambda}U(h))=h(1)\int_{\lambda\in
C_{S}\atop|\lambda|\in[\Lambda^{-1},\Lambda]}d^{*}\lambda+\sum_{\nu\in
S}\int^{\prime}_{{\mathbb{Q}}_{\nu}^{*}}\frac{h(u^{-1})}{|1-u|}d^{*}u+o(1)\qquad\text{for
$\Lambda\to\infty$}$

(later recast in the softer context of [13]). There is a rather simple analogy
related to the spectral (left-hand) side of the explicit formulas for a global
field $K$ (see [4] Section 2 for the notations)

$\hat{h}(0)+\hat{h}(1)-\sum_{\chi\in\widehat{C_{K,1}}}\sum_{\rho\in
Z_{\tilde{\chi}}}\hat{h}(\tilde{\chi},\rho)=\sum_{\nu}\int^{\prime}_{K_{v}^{*}}\frac{h(u^{-1})}{|1-u|}d^{*}u$

which may help one to realize how the sum over the zeros of the zeta function
appears. Here this relation is simply explained. Given a complex valued
polynomial $P(x)\in{\mathbb{C}}[x]$, one may identify the set of its zeros as
the spectrum of the endomorphism $T$ of multiplication by the variable $x$
computed in the quotient algebra ${\mathbb{C}}[x]/(P(x))$. It is well known
that the matrix of $T$, in the basis of powers of $x$, is the companion matrix
of $P(x)$. Furthermore, the trace of its powers, readily computed from the
diagonal terms of powers of the companion matrix in terms of the coefficients
of $P(x)$, gives the Newton-Girard formulae.111This is an efficient way to
find the power sum of roots of $P(x)$ without actually finding the roots
explicitly. Newton’s identities supply the calculation via a recurrence
relation with known coefficients. If one transposes this result to the case of
the Riemann zeta function $\zeta(s)$, one sees that the multiplication by
$P(x)$ is replaced here with the map

(2.1) ${\mathcal{E}}(f)(u):=u^{1/2}\sum_{n=1}^{\infty}f(nu),$

while the role of $T$ (the multiplication by the variable) is played by the
scaling operator $u\partial_{u}$. These statements may become more evident if
one brings in the Fourier transform. Indeed, let
$f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ be an even Schwartz function and
let $w(f)(u)=u^{1/2}f(u)$ be the unitary identification of $f$ with a function
in $L^{2}({\mathbb{R}}_{+}^{*},d^{*}u)$, where $d^{*}u:=du/u$ denotes the Haar
measure. Then, by composing $w$ with the (multiplicative) Fourier transform
${\mathbb{F}}:L^{2}({\mathbb{R}}_{+}^{*},d^{*}u)\to L^{2}({\mathbb{R}})$,
${\mathbb{F}}(h)(s)=\int_{{\mathbb{R}}_{+}^{*}}h(u)u^{-is}d^{*}u$ one obtains

${\mathbb{F}}(w(f))=\psi,\qquad\psi(z)=\int_{{\mathbb{R}}_{+}^{*}}f(u)u^{\frac{1}{2}-iz}d^{*}u.$

The function $\psi(z)$ is holomorphic in the complex half-plane
$\mathfrak{H}=\\{z\in{\mathbb{C}}~{}|~{}\Im(z)>-\frac{1}{2}\\}$ since
$f(u)=O(u^{-N})$ for $u\to\infty$. Moreover, for $n\in{\mathbb{N}}$, one has

$\int_{{\mathbb{R}}_{+}^{*}}u^{1/2}f(nu)u^{-iz}d^{*}u=n^{-1/2+iz}\int_{{\mathbb{R}}_{+}^{*}}v^{1/2}f(v)v^{-iz}d^{*}v.$

In the region $\Im(z)>\frac{1}{2}$ one derives, by applying Fubini theorem,
the following equality

$\int_{{\mathbb{R}}_{+}^{*}}{\sum_{n=1}^{\infty}}u^{1/2}f(nu)u^{-iz}d^{*}u=\left({\sum_{n=1}^{\infty}}n^{-1/2+iz}\right)\int_{{\mathbb{R}}_{+}^{*}}v^{1/2}f(v)v^{-iz}d^{*}v.$

Thus, for all $z\in{\mathbb{C}}$ with $\Im(z)>\frac{1}{2}$ one obtains

(2.2)
$\int_{{\mathbb{R}}_{+}^{*}}{\mathcal{E}}(f)(u)u^{-iz}d^{*}u=\zeta(\frac{1}{2}-iz)\psi(z).$

If one assumes now that the Schwartz function $f$ fulfills
$\int_{\mathbb{R}}f(x)dx=0$, then $\psi(\frac{i}{2})=0$. Both sides of (2.2)
are holomorphic functions in $\mathfrak{H}$: for the integral on the left-hand
side, this can be seen by using the estimate ${\mathcal{E}}(f)(u)=O(u^{1/2})$
that follows from the Poisson formula. This proves that (2.2) continues to
hold also in the complex half-plane $\mathfrak{H}$. Thus one sees that the
zeros of $\zeta(\frac{1}{2}-iz)$ in the strip $|\Im(z)|<\frac{1}{2}$ are the
common zeros of all functions

${\mathbb{F}}({\mathcal{E}}(f))(z),\qquad f\in{\mathcal{S}}({\mathbb{R}})^{\rm
ev}_{1}:=\\{f\in{\mathcal{S}}({\mathbb{R}})^{\rm
ev}~{}|~{}\int_{\mathbb{R}}f(x)dx=0\\}.$

One may eventually select the even Schwartz function $f(x)=e^{-\pi
x^{2}}\left(2\pi x^{2}-1\right)$ to produce a specific instance where the
zeros of ${\mathbb{F}}({\mathcal{E}}(f))$ are exactly the non-trivial zeros of
$\zeta(\frac{1}{2}-iz)$, since in this case
$\psi(z)=\frac{1}{4}\pi^{-\frac{1}{4}+\frac{iz}{2}}(-1-2iz)\Gamma\left(\frac{1}{4}-\frac{iz}{2}\right)$.

##  3\. Geometric interpretation

In this section we continue the study of the map ${\mathcal{E}}$ with the goal
to achieve a geometric understanding of it. This is obtained by bringing in
the construction the adèle class space of the rationals, whose role is that to
grant for the replacement, in (2.1), of the summation over the monoid
${\mathbb{N}}^{\times}$ with the summation over the group
${\mathbb{Q}}^{\times}$. Then, up to the factor $u^{1/2}$, ${\mathcal{E}}$ is
understood as the composite $\iota^{*}\circ\pi_{!}$, where the map
$\iota:{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}^{*}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}$
is the inclusion of idèle classes in adèle classes and
$\pi:{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}$
is induced by the projection
${\mathbb{A}}_{\mathbb{Q}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$.
We shall discuss the following diagram

$\textstyle{Y={\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}}$$\textstyle{{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}^{\times}_{\mathbb{Q}}/{\hat{\mathbb{Z}}^{\times}}={\mathbb{R}}_{+}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$

The conceptual understanding of the map $\pi_{!}$ uses Hochschild homology of
noncommutative algebras.

We recall that the space of adèle classes i.e. the quotient
${\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}$ is encoded
algebraically by the cross-product algebra

${\mathcal{A}}:={\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\rtimes{\mathbb{Q}}^{\times}.$

The Schwartz space ${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ is acted upon by
(automorphisms of) ${\mathbb{Q}}^{\times}$ corresponding to the scaling action
of ${\mathbb{Q}}^{\times}$ on rational adèles. An element of ${\mathcal{A}}$
is written symbolically as a finite sum

$\sum a(q)U(q),\qquad a(q)\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}}).$

From the inclusion of algebras
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\subset{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\rtimes{\mathbb{Q}}^{\times}={\mathcal{A}}$
one derives a corresponding morphism of Hochschild homologies

$\pi_{!}:\text{HH}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}}))\longrightarrow\text{HH}({\mathcal{A}}).$

Here, we use the shorthand notation $\text{HH}(A):=\text{HH}(A,A)$ for the
Hochschild homology of an algebra $A$ with coefficients in the bimodule $A$.
In noncommutative geometry, the vector space of differential forms of degree
$k$ is replaced by the Hochschild homology $\text{HH}_{k}(A)$. If the algebra
$A$ is commutative and for $k=0$, $\text{HH}_{0}(A)=A$, so that $0$-forms are
identified with functions. Indeed, the Hochschild boundary map

$b:A^{\otimes\,2}\to A,\qquad b(x\otimes y)=xy-yx$

is identically zero when the algebra $A$ is commutative. This result does not
hold when $A={\mathcal{A}}$, since
${\mathcal{A}}={\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\rtimes{\mathbb{Q}}^{\times}$
is no longer commutative. It is therefore meaningful to bring in the following

######  Proposition 3.1.

The kernel of
$\pi_{!}:\text{HH}_{0}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}}))\to\text{HH}_{0}({\mathcal{A}})$
is the ${\mathbb{C}}$-linear span $E$ of functions $f-f_{q}$, with
$f\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$, $q\in{\mathbb{Q}}^{\times}$,
and where we set $f_{q}(x):=f(qx)$.

###### Proof.

For any $f,g\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ and
$q\in{\mathbb{Q}}^{\times}$ one has

(3.1) $fg-(fg)_{q}=fg-U(q)fgU(q^{-1})=xy-yx,\qquad x:=fU(q^{-1}),\ y:=U(q)g.$

One knows ([14] Lemma 1) that any function $f\in{\mathcal{S}}({\mathbb{R}})$
is a product of two elements of ${\mathcal{S}}({\mathbb{R}})$. Moreover, an
element of the Bruhat-Schwartz space
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ is a finite linear combination of
functions of the form $e\otimes f$, with $e^{2}=e$. Thus any
$f\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ can be written as a finite sum
of products of two elements of ${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$, so
that (3.1) entails $f-f_{q}\in\ker\pi_{!}$. Conversely, let $f\in\ker\pi_{!}$.
Then there exists a finite number of pairs $x_{i},y_{i}\in{\mathcal{A}}$ such
that $f=\sum[x_{i},y_{i}]$. Let
$P:{\mathcal{A}}\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ be the projection
on the coefficient $a(1)$ of $U(1)=1$ i.e.

$P\left(\sum a(q)U(q)\right):=a(1).$

Then $f=\sum[x_{i},y_{i}]$ implies $f=\sum P([x_{i},y_{i}])$. We shall prove
that for any pair $x,y\in{\mathcal{A}}$ one has $P([x,y])\in E$. Indeed, one
has

$x=\sum a(q)U(q),\ y=\sum b(q^{\prime})U(q^{\prime}),\
[x,y]=\sum(a(q)b(q^{\prime})_{q}-b(q^{\prime})a(q)_{q^{\prime}})U(qq^{\prime})$

so that

$P([x,y])=\sum(a(q)b(q^{-1})_{q}-b(q^{-1})a(q)_{q^{-1}}).$

This projection belongs to $E$ in view of the fact that

$a(q)b(q^{-1})_{q}-b(q^{-1})a(q)_{q^{-1}}=h_{q}-h,\qquad
h=b(q^{-1})a(q)_{q^{-1}}.$

This completes the proof.∎

Proposition 3.1 shows that the image of
$\pi_{!}:\text{HH}_{0}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}}))\to\text{HH}_{0}({\mathcal{A}})$
is the space of coinvariants for the action of ${\mathbb{Q}}^{\times}$ on
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$, i.e. the quotient of
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ by the subspace $E$.
An important point now to remember is that the Fourier transform becomes
canonically defined on the above quotient. Indeed, the definition of the
Fourier transform on adèles depends on the choice of a non-trivial character
$\alpha$ on the additive, locally compact group ${\mathbb{A}}_{\mathbb{Q}}$,
which is trivial on the subgroup
${\mathbb{Q}}\subset{\mathbb{A}}_{\mathbb{Q}}$. It is defined as follows

${\mathbb{F}}_{\alpha}(f)(y):=\int_{\mathbb{R}}f(x)\alpha(xy)dx.$

The space of characters of the compact group
$G={\mathbb{A}}_{\mathbb{Q}}/{\mathbb{Q}}$ is one dimensional as a
${\mathbb{Q}}$-vector space, thus any non-trivial character $\alpha$ as above
is of the form $\beta(x)=\alpha(qx)$, so that

${\mathbb{F}}_{\beta}(f)(y):=\int_{\mathbb{R}}f(x)\alpha(qxy)dx={\mathbb{F}}_{\alpha}(f)_{q}(y).$

Therefore, the difference ${\mathbb{F}}_{\beta}-{\mathbb{F}}_{\alpha}$
vanishes on the quotient of ${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ by $E$
and this latter space is preserved by ${\mathbb{F}}_{\alpha}$ since
${\mathbb{F}}_{\alpha}(f_{q})={\mathbb{F}}_{\alpha}(f)_{q^{-1}}$.

###  3.1. HH, Morita invariance and the trace map

Let us recall that given an algebra $A$, the trace map

${\rm Tr}:M_{n}(A)\to A,\qquad{\rm Tr}((a_{ij}):=\sum a_{jj}$

induces an isomorphism in degree zero Hochschild homology which extends to
higher degrees. If $A$ is a convolution algebra of the étale groupoid of an
equivalence relation ${\mathcal{R}}$ with countable orbits on a space $Y$, and
$\pi:Y\to Y/{\mathcal{R}}$ is the quotient map, the trace map takes the
following form

(3.2) ${\rm Tr}(f)(x):=\sum_{\pi(j)=x}f(j,j).$

The trace induces a map on $\text{HH}_{0}$ of the function algebras, provided
one takes care of the convergence issue when the size of equivalence classes
is infinite. If the relation ${\mathcal{R}}$ is associated with the orbits of
the free action of a discrete group $\Gamma$ on a locally compact space $Y$,
the convolution algebra is the cross product of the algebra of functions on
$Y$ by the discrete group $\Gamma$. In this case, the étale groupoid is
$Y\rtimes\Gamma$, where the source and range maps are given resp. by
$s(y,g)=y$ and $r(y,g)=gy$. The elements of the convolution algebra are
functions $f(y,g)$ on $Y\rtimes\Gamma$. The diagonal terms in (3.2) correspond
to the elements of $Y\rtimes\Gamma$ such that $s(y,g)=r(y,g)$, meaning that
$g=1$ is the neutral element of $\Gamma$, since the action of $\Gamma$ is
assumed to be free. Then, the trace map is

${\rm Tr}((f)(x)=\sum_{\pi(y)=x}f(y,1).$

This sum is meaningful on the space of the proper orbits of $\Gamma$. For a
lift $\rho(x)\in Y$, with $\pi(\rho(x))=x$ the trace reads as

(3.3) ${\rm Tr}(f)(x)=\sum_{g\in\Gamma}f(g\rho(x),1).$

In the case of $Y={\mathbb{A}}_{\mathbb{Q}}$ acted upon by
$\Gamma={\mathbb{Q}}^{\times}$, the proper orbits are parameterized by the
idèle classes and this space embeds in the adèle classes by means of the
inclusion

$\iota:{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}^{\times}_{\mathbb{Q}}\to{\mathbb{Q}}^{\times}\backslash{\mathbb{A}}_{\mathbb{Q}}.$

We identify the idèle class group
$C_{\mathbb{Q}}={\mathbb{Q}}^{\times}\backslash{\mathbb{A}}^{\times}_{\mathbb{Q}}$
with ${\hat{\mathbb{Z}}^{\times}}\times{\mathbb{R}}_{+}^{*}$, using the
canonical exact sequence affected by the modulus

$1\to{\hat{\mathbb{Z}}^{\times}}\to C_{\mathbb{Q}}\stackrel{{\scriptstyle{\rm
Mod}}}{{\longrightarrow}}{\mathbb{R}}_{+}^{*}\to 1.$

There is a natural section
$\rho:C_{\mathbb{Q}}\to{\mathbb{A}}^{\times}_{\mathbb{Q}}$ of the quotient
map, given by the canonical inclusion

${\hat{\mathbb{Z}}^{\times}}\times{\mathbb{R}}_{+}^{*}\subset{\mathbb{A}}_{\mathbb{Q}}^{f}\times{\mathbb{R}}={\mathbb{A}}_{\mathbb{Q}}.$

Next, we focus on the ${\hat{\mathbb{Z}}^{\times}}$-invariant part of
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$. Then, with the notations of
Proposition 3.1 we have

######  Lemma 3.2.

The following facts hold

  1. (i)

Let
$h\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}$,
then there exists $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ with
$h-1_{\hat{\mathbb{Z}}}\otimes f\in E^{{\hat{\mathbb{Z}}^{\times}}}$.

  2. (ii)

Let $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ and
$\tilde{f}=(1_{\hat{\mathbb{Z}}}\otimes f)U(1)\in
S({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}\rtimes{\mathbb{Q}}^{\times}$,
then one has

(3.4) ${\rm
Tr}(\tilde{f})(u)=2\sum_{n\in{\mathbb{N}}^{\times}}f(nu)\qquad\forall
u\in{\mathbb{R}}_{+}^{*}.$

  3. (iii)

Let $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$, then
$1_{\hat{\mathbb{Z}}}\otimes f\in E^{{\hat{\mathbb{Z}}^{\times}}}\iff f=0$.

###### Proof.

(i) By definition, the elements of the Bruhat-Schwartz space
${{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})}$ are finite linear combinations of
functions on ${\mathbb{A}}_{\mathbb{Q}}$ of the form ($S\ni\infty$ is a finite
set of places)

$f=\otimes f_{v},\qquad f_{v}=1_{{\mathbb{Z}}_{v}}~{}\forall v\notin S,\quad
f_{\infty}\in{\mathcal{S}}({\mathbb{R}}),\quad
f_{p}\in{\mathcal{S}}({\mathbb{Q}}_{p})\quad\forall p\in S\setminus\infty,$

where ${\mathcal{S}}({\mathbb{Q}}_{p})$ denotes the space of locally constant
functions with compact support. An element of
${\mathcal{S}}({\mathbb{Q}}_{p})$ which is ${\mathbb{Z}}_{p}^{*}$-invariant is
a finite linear combination of characteristic functions
$(1_{{\mathbb{Z}}_{p}})_{p^{n}}(x):=1_{{\mathbb{Z}}_{p}}(p^{n}x)$. Thus an
element
$h\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}$
is a finite linear combination of functions of the form

$f=\otimes f_{v},\qquad f_{v}=1_{{\mathbb{Z}}_{v}}\quad\forall v\notin S,\quad
f_{\infty}\in{\mathcal{S}}({\mathbb{R}}),\quad
f_{p}=(1_{{\mathbb{Z}}_{p}})_{p^{n_{p}}}\quad\forall p\in S\setminus\infty$

With $q=\prod p^{-n_{p}}$ one has with $\ell(x):=f(qx)$,
$\ell=1_{\hat{\mathbb{Z}}}\otimes g$, $\ell-f\in
E^{{\hat{\mathbb{Z}}^{\times}}}$ and the replacement of $g$ with its even part
$\frac{1}{2}(g(x)+g(-x))$ does not change the class of $f$ modulo
$E^{{\hat{\mathbb{Z}}^{\times}}}$.
(ii) Let $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ and
$\tilde{f}=(1_{\hat{\mathbb{Z}}}\otimes f)U(1)\in
S({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}\rtimes{\mathbb{Q}}^{\times}$.
For $u\in{\mathbb{R}}_{+}^{*}$ the lift
$\rho(u)\in{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$ is
$(1,u)\in{\hat{\mathbb{Z}}^{\times}}\times{\mathbb{R}}_{+}^{*}\subset{\mathbb{A}}_{\mathbb{Q}}^{f}\times{\mathbb{R}}={\mathbb{A}}_{\mathbb{Q}}$,
and by applying (3.3) one has

(3.5) ${\rm
Tr}(\tilde{f})(u)=\sum_{q\in{\mathbb{Q}}^{\times}}\tilde{f}(q\rho(x),1)=\sum_{q\in{\mathbb{Q}}^{\times}}(1_{\hat{\mathbb{Z}}}\otimes
f)(q,qu)=2\sum_{n\in{\mathbb{N}}^{\times}}f(nu)\quad\forall
u\in{\mathbb{R}}_{+}^{*}.$

(iii) Let $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ such that
$1_{\hat{\mathbb{Z}}}\otimes f\in E^{{\hat{\mathbb{Z}}^{\times}}}$. Let
$\tilde{f}=(1_{\hat{\mathbb{Z}}}\otimes f)U(1)\in
S({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}\rtimes{\mathbb{Q}}^{\times}$.
By Proposition 3.1 the Hochschild class in $\text{HH}_{0}({\mathcal{A}})$ of
$\tilde{f}$ is zero, thus ${\rm Tr}(\tilde{f})=0$. It follows from (3.4) that
${\mathcal{E}}(f)(u)=0$ $\forall u\in{\mathbb{R}}_{+}^{*}$. Then (2.2) implies
that the function
$\psi(z)=\int_{{\mathbb{R}}_{+}^{*}}f(u)u^{\frac{1}{2}-iz}d^{*}u$ is well
defined in the half-plane $\Im(z)>\frac{1}{2}$ where it vanishes identically,
thus $f=0$. The converse of the statement is obvious. ∎

The next statement complements Proposition 3.1, with a description of the
range of
$\pi_{!}:\text{HH}_{0}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}})\to\text{HH}_{0}({\mathcal{A}})^{{\hat{\mathbb{Z}}^{\times}}}$;
it also shows that the map
${\mathcal{E}}(f)(u)=u^{1/2}\sum_{n=1}^{\infty}f(nu)$ coincides, up to the
factor $\frac{u^{1/2}}{2}$, with the trace map (3.5). We keep the notations of
Lemma 3.2

######  Proposition 3.3.

The map ${\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to\pi_{!}\left(\text{HH}_{0}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}})\right)$,
$f\mapsto\tilde{f}$ is an isomorphism, this means that
$\pi_{!}\left(\text{HH}_{0}({\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}})\right)$
is determined by the images of the elements of the subalgebra
$1_{\hat{\mathbb{Z}}}\otimes{\mathcal{S}}({\mathbb{R}})^{\rm
ev}\subset{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}$.
Furthermore, one has the identity

${\mathcal{E}}(f)(u)=\frac{u^{1/2}}{2}\ {\rm Tr}(\tilde{f})(u)\quad
f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev},~{}\forall u\in{\mathbb{R}}_{+}^{*}.$

###### Proof.

The first statement follows from Lemma 3.2 (i) and (iii). The second statement
from (ii) of the same lemma. ∎

##  4\. The Laplacian $\Delta=H(1+H)$

This section describes the spectral interpretation of the squares of non-
trivial zeros of the Riemann zeta function in terms of a suitable Laplacian.
It also shows the relation between this Laplacian and the prolate wave
operator.

###  4.1. The vanishing conditions

One starts with the exact sequence

$0\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})_{1}\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\stackrel{{\scriptstyle\epsilon}}{{\to}}{\mathbb{C}}(1)\to
0$

associated to the kernel of the ${\mathbb{Q}}^{\times}$-invariant linear
functional
$\epsilon(f)=\int_{{\mathbb{A}}_{\mathbb{Q}}}f(x)dx\in{\mathbb{C}}(1)$. By
implementing in the above sequence the evaluation $\delta_{0}(f):=f(0)$, one
obtains the exact sequence

$0\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})_{0}\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\stackrel{{\scriptstyle(\delta_{0},\epsilon)}}{{\longrightarrow}}{\mathbb{C}}(0)\oplus{\mathbb{C}}(1)\to
0.$

The next lemma shows that both ${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})_{0}$
and ${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})_{1}$ have a description in terms
of the ranges of two related differential operators. For simplicity of
exposition, we restrict our discussion to the
${\hat{\mathbb{Z}}^{\times}}$-invariant parts of these function spaces.

######  Lemma 4.1.

Let
$H:{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})\to{\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})$,
$H:=x\partial_{x}$ be the generator of the scaling action of
$1\times{\mathbb{R}}_{+}^{*}\subset{\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$.
Then one has

  1. (i)

$H$ commutes with the action of ${\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$ and
restricts to
${\mathcal{S}}({\mathbb{A}}_{\mathbb{Q}})^{{\hat{\mathbb{Z}}^{\times}}}$.

  2. (ii)

$(1+H)$ induces an isomorphism ${\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to{\mathcal{S}}({\mathbb{R}})_{1}^{\rm ev}$.

  3. (iii)

$H(1+H)$ induces an isomorphism ${\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to{\mathcal{S}}({\mathbb{R}})_{0}^{\rm ev}$.

###### Proof.

(i) follows since ${\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$ is abelian, thus
$H$ commutes with the action of ${\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$.
(ii) The functional
$f\mapsto\epsilon(f)=\int_{{\mathbb{A}}_{\mathbb{Q}}}f(x)dx$ vanishes on the
range of $1+H$ since

$\epsilon((1+H)f)=\int_{{\mathbb{A}}_{\mathbb{Q}}}\partial(xf)dx=0,$

thus the range of $(1+H)$ is contained in ${\mathcal{S}}({\mathbb{R}})^{\rm
ev}_{1}$. The equation $Hf+f=0$ implies that $xf(x)$ is constant, hence $f=0$,
for $f\in{\mathcal{S}}({\mathbb{R}})$. Thus
$(1+H):{\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to{\mathcal{S}}({\mathbb{R}})_{1}^{\rm ev}$ is injective. Let now
$f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}_{1}$ and let $\widehat{f}$ be its
Fourier transform. Then $\widehat{f}\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$
and $\widehat{f}(0)=0$. The function $g(x):=\widehat{f}(x)/x$, $g(0):=0$ is
smooth, and $g\in{\mathcal{S}}({\mathbb{R}})^{\rm odd}$, while the function

$h(x):=\int_{-\infty}^{x}g(y)dy$

fulfills $h\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ and $\partial_{x}h=g$. One
has $Hh=\widehat{f}$, thus $(-1-H)\widehat{h}=f$. This shows that
$(1+H):{\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to{\mathcal{S}}({\mathbb{R}})_{1}^{\rm ev}$ is also surjective.
(iii) Since the evaluation $f\mapsto f(0)$ is invariant under scaling, it
vanishes on the range of $H$. Similarly, one sees that the functional
$f\mapsto\int_{{\mathbb{A}}_{\mathbb{Q}}}f(x)dx$ vanishes on the range of
$1+H$. Thus the range of $H(1+H)$ is contained in
${\mathcal{S}}({\mathbb{R}})_{0}$. The equation $Hf=0$ implies that the
function $f$ is constant and hence $f=0$, for
$f\in{\mathcal{S}}({\mathbb{R}})$. Similarly $Hf+f=0$ implies that $xf(x)$ is
constant and hence $f=0$ for $f\in{\mathcal{S}}({\mathbb{R}})$. Thus
$H(1+H):{\mathcal{S}}({\mathbb{R}})\to{\mathcal{S}}({\mathbb{R}})_{0}$ is
injective. Let now $f\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ with $f(0)=0$.
Then the function $g(x):=f(x)/x$, $g(0):=0$, is smooth,
$g\in{\mathcal{S}}({\mathbb{R}})^{\rm odd}$ and there exists a unique
$h\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ such that $\partial_{x}h=g$. One
has $Hh=f$ so that $(-1-H)\widehat{h}=\widehat{f}$. Thus if $\widehat{f}(0)=0$
one has $\widehat{h}(0)=0$ and there exists
$k\in{\mathcal{S}}({\mathbb{R}})^{\rm ev}$ with $Hk=\widehat{h}$. Then
$-(1+H)\widehat{k}=h$ and $H(1+H)\widehat{k}=-f$. This shows that
$H(1+H):{\mathcal{S}}({\mathbb{R}})^{\rm
ev}\to{\mathcal{S}}({\mathbb{R}})_{0}^{\rm ev}$ is surjective and an
isomorphism.∎

###  4.2. The Laplacian $\Delta=H(1+H)$ and its spectrum

This section is based on the following heuristic dictionary suggesting a
parallel between some classical notions in Hodge theory on the left-hand side,
and their counterparts in noncommutative geometry, for the adèle class space
of the rationals. The notations are inclusive of those of Section 3

Algebra of functions |  Cross-product by ${\mathbb{Q}}^{\times}$
---|---
Differential forms | Hochschild homology
Star operator $\star$ |  $\iota\ \times$ ${\mathbb{F}}$
Differential $d$ |  Operator $H$
$\delta:=\star d\star$ |  Operator $1+H$
Laplacian | $\Delta:=H(1+H)$

Next Proposition is a variant of the spectral realization in [8, 9].

######  Proposition 4.2.

The following facts hold

  1. (i)

The trace map ${\rm Tr}$ commutes with $\Delta=H(1+H)$ and the range of ${\rm
Tr}\circ\Delta$ is contained in the strong Schwartz space ${\bf
S\,}({\mathbb{R}}_{+}^{*}):=\,\cap_{\beta\in{\mathbb{R}}}\,\mu^{\beta}{\mathcal{S}}({\mathbb{R}}_{+}^{*})$,
with $\mu$ denoting the Modulus.

  2. (ii)

The spectrum of $\Delta$ on the quotient of ${\bf S\,}({\mathbb{R}}_{+}^{*})$
by the closure of the range of ${\rm Tr}\circ\Delta$ is the set (counted with
possible multiplicities)

$\left\\{\Big{(}z-\frac{1}{2}\Big{)}^{2}-\frac{1}{4}\mid
z\in{\mathbb{C}}\setminus{\mathbb{R}},~{}\zeta(z)=0\right\\}.$

###### Proof.

(i) The trace map of (3.5) commutes with $\Delta$. By Lemma 4.1 (iii) the
range of $\Delta$ is ${\mathcal{S}}({\mathbb{R}})_{0}^{\rm ev}$ thus the range
of ${\mathcal{E}}\circ(H(1+H))$ is contained in ${\bf
S\,}({\mathbb{R}}_{+}^{*})$ (see [9], Lemma 2.51).
(ii) By construction, ${\bf S\,}({\mathbb{R}}_{+}^{*})$ is the intersection,
indexed by compact intervals $J\subset{\mathbb{R}}$, of the spaces
$\cap_{\beta\in J}\,\mu^{\beta}{\mathcal{S}}({\mathbb{R}}_{+}^{*})$. The
Fourier transform

${\mathbb{F}}(f)(z):=\int_{{\mathbb{R}}_{+}^{*}}f(u)u^{z}d^{*}u$

defines an isomorphism of these function spaces with the Schwartz spaces
${\mathcal{S}}(I)$, labeled by vertical strips
$I:=\\{z\in{\mathbb{C}}\mid\Re(z)\in J\\}$, of holomorphic functions $f$ in
$I$ with $p_{k,m}(f)<\infty$ for all $k,m\in{\mathbb{N}}$ where

$p_{k,m}(f)=\sum_{n=0}^{m}\,\frac{1}{n!}\,\sup_{I}(1+|z|)^{k}\cdot|\partial^{n}f(z)|\,.$

These norms define the Fréchet topology on ${\mathcal{S}}(I)$.
It then follows from Lemma 4.125 of [9] that, for $I$ sufficiently large, the
quotient of ${\mathcal{S}}(I)$ by the closure of the range of ${\rm
Tr}\circ\Delta$ decomposes into a direct sum of finite-dimensional spaces
associated to the projections $\Pi(N)$, $N\in{\mathbb{Z}}$ which fulfill the
following properties

  1. (1)

Each $\Pi(N)$ is an idempotent.

  2. (2)

The sequence of $\Pi(N)$’s is of tempered growth.

  3. (3)

The rank of $\Pi(N)$ is $O(\log|N|)$ for $|N|\to\infty$.

  4. (4)

For any $f\in{\mathcal{S}}(I)$ the sequence $f\,\Pi(N)$ is of rapid decay and

$\sum_{N\in{\mathbb{Z}}}\,f\,\Pi(N)=f\qquad\forall f\in{\mathcal{S}}(I)\,.$

This direct sum decomposition commutes with $\Delta$ since both $\Pi(N)$ and
the conjugate of $\Delta$ by the Fourier transform ${\mathbb{F}}$ are given by
multiplication operators. The conjugate of $H$ by ${\mathbb{F}}$ is the
multiplication by $-z$, so that the conjugate of $\Delta$ is the
multiplication by $-z(1-z)$. The spectrum of $\Delta$ is the union of the
spectra of the finite-dimensional operators
$\Delta_{N}:=\Pi(N)\Delta=\Delta\Pi(N)$. By [9], Corollary 4.118, and the
proof of Theorem 4.116, the finite-dimensional range of $\Pi(N)$ is described
by the evaluation of $f\in{\mathcal{S}}(I)$ on the zeros $\rho\in Z(N)$ of the
Riemann zeta function which are inside the contour $\gamma_{N}$, i.e. by the
map

${\mathcal{S}}(I)\ni f\mapsto f|_{Z(N)}=(f^{(n_{\rho})}(\rho))_{\rho\in
Z(N)}\in{\mathbb{C}}^{(n_{\rho})}$

where ${\mathbb{C}}^{(n_{\rho})}$ denotes the space of dimension $n_{\rho}$ of
jets of order equal to the order $n_{\rho}$ of the zero $\rho$ of the zeta
function. Moreover, the action of $\Delta_{N}$ is given by the matrix
associated with the multiplication of $f\in{\mathcal{S}}(I)$ by $-z(1-z)$:
this gives a triangular matrix whose diagonal is given by $n_{\rho}$ terms all
equal to $-\rho(1-\rho)$. Thus the spectrum of $\Delta$ on the quotient of
${\bf S\,}({\mathbb{R}}_{+}^{*})$ by the closure of the range of ${\rm
Tr}\circ\Delta$ is the set (counted with multiplicities)

$\left\\{\Big{(}\rho-\frac{1}{2}\Big{)}^{2}-\frac{1}{4}\mid\rho\in{\mathbb{C}}\setminus{\mathbb{R}},\
\zeta(\rho)=0\right\\}.$

∎

######  Corollary 4.3.

The spectrum of $\Delta$ on the quotient of the strong Schwartz space ${\bf
S\,}({\mathbb{R}}_{+}^{*})$ by the closure of the range of ${\rm
Tr}\circ\Delta$ is negative if and only if the Riemann Hypothesis holds.

###### Proof.

This follows from Proposition 4.2 and the fact that for $\rho\in{\mathbb{C}}$

$\Big{(}\rho-\frac{1}{2}\Big{)}^{2}-\frac{1}{4}\leq
0\iff\rho\in[0,1]\cup\frac{1}{2}+i{\mathbb{R}}.$

∎

######  Remark 4.4.

The main interest of the above reformulation of the spectral realization of
[8, 9] in terms of the Laplacian $\Delta$ is that the latter is intimately
related to the prolate wave operator $W_{\lambda}$ that is shown in [10] to be
self-adjoint and have, for $\lambda=\sqrt{2}$ the same UV spectrum as the
Riemann zeta function. The relation between $\Delta$ and $W_{\lambda}$ is that
the latter is a perturbation of $\Delta$ by a multiple of the Harmonic
oscillator.

##  5\. Sheaves on the Scaling Site and
$H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}})$

Let $\mu\in{\mathbb{R}}_{>1}$ and $\Sigma_{\mu}$ be the linear map on
functions $g:{\mathbb{R}}^{*}_{+}\to{\mathbb{C}}$ of sufficiently rapid decay
at $0$ and $\infty$ defined by

(5.1) $(\Sigma_{\mu}g)(u)=\sum_{k\in{\mathbb{Z}}}g(\mu^{k}u).$

We shall denote with ${{\mathcal{S}}^{\rm ev}_{0}}$ the linear space of real-
valued, even Schwartz functions $f\in{\mathcal{S}}({\mathbb{R}})$ fulfilling
the two conditions $f(0)=0=\int_{\mathbb{R}}f(x)dx$. The map

(5.2) ${\mathcal{E}}:{{\mathcal{S}}^{\rm
ev}_{0}}\to{\mathbb{R}},\qquad({\mathcal{E}}f)(u)=u^{1/2}\sum_{n>0}f(nu)$

is proportional to a Riemann sum for the integral of $f$. The following lemma
on scale invariant Riemann sums justifies the pointwise “well-behavior” of
(5.2) (see [7] Lemma 6.1)

######  Lemma 5.1.

Let $f$ be a complex-valued function of bounded variation on $(0,\infty)$.
Assume that $f$ is of rapid decay for $u\to\infty$, $O(u^{2})$ when $u\to 0$,
and that $\int_{0}^{\infty}f(t)dt=0$. Then the following properties hold

  1. (i)

The function $({\mathcal{E}}f)(u)$ in (5.2) is well-defined pointwise, is
$O(u^{1/2})$ when $u\to 0$, and of rapid decay for $u\to\infty$.

  2. (ii)

Let $g={\mathcal{E}}(f)$, then the series (5.1) is geometrically convergent,
and defines a bounded and measurable function on
${\mathbb{R}}_{+}^{*}/\mu^{{\mathbb{Z}}}$.

We recall that a sheaf over the Scaling Site
${\mathscr{S}}={[0,\infty)\rtimes{{\mathbb{N}}^{\times}}}$ is a sheaf of sets
on $[0,\infty)$ (endowed with the euclidean topology) which is equivariant for
the action of the multiplicative monoid ${\mathbb{N}}^{\times}$ [6]. Since we
work in characteristic zero we select as structure sheaf of ${\mathscr{S}}$
the ${\mathbb{N}}^{\times}$-equivariant sheaf ${\mathcal{O}}$ whose sections
on an open set $U\subset[0,\infty)$ define the space of smooth, complex-valued
functions on $U$. The next proposition introduces two relevant sheaves of
${\mathcal{O}}$-modules.

######  Proposition 5.2.

Let $L\in(0,\infty)$, $\mu=\exp L$, and
$C_{\mu}={\mathbb{R}}^{*}_{+}/\mu^{\mathbb{Z}}$. The following facts hold

  1. (i)

As $L$ varies in $(0,\infty)$, the pointwise multiplicative Fourier transform

(5.3)
${\mathbb{F}}:L^{2}(C_{\mu})\to\ell^{2}({\mathbb{Z}}),\qquad{\mathbb{F}}(\xi)(n)=L^{-\frac{1}{2}}\int_{C_{\mu}}\xi(u)u^{-\frac{2\pi
in}{L}}d^{*}u$

defines an isomorphism between the family of Hilbert spaces $L^{2}(C_{\mu})$
and the restriction to $(0,\infty)$ of the trivial vector bundle
$L^{2}=[0,\infty)\times\ell^{2}({\mathbb{Z}})$.

  2. (ii)

The smooth sections vanishing at $L=0$ of the vector bundle $L^{2}$ together
with the linear maps

(5.4) $\sigma_{n}:L^{2}(C_{\mu^{n}})\to
L^{2}(C_{\mu}),\qquad\sigma_{n}(\xi)(u)=\sum_{j=1}^{n}\xi(\mu^{j}u)$

define a sheaf ${\mathcal{L}}^{2}$ of ${\mathcal{O}}$-modules on
${\mathscr{S}}$.

  3. (iii)

For $f\in{{\mathcal{S}}^{\rm ev}_{0}}$, the maps
$\Sigma{\mathcal{E}}(f)(L):=\Sigma_{\mu}{\mathcal{E}}(f)$ as in (5.1) define
smooth (global) sections $\Sigma{\mathcal{E}}(f)$ of ${\mathcal{L}}^{2}$.

  4. (iv)

For any open set $U\subset[0,\infty)$, the submodules closure of
$C_{0}^{\infty}(U,\Sigma{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}}))$ in
$C_{0}^{\infty}(U,L^{2})$ are ${\mathbb{N}}^{\times}$-equivariant and define a
subsheaf $\overline{\Sigma{\mathcal{E}}}\subset{\mathcal{L}}^{2}$ of closed
${\mathcal{O}}$-modules on ${\mathscr{S}}$.

  5. (v)

For $U\subset[0,\infty)$ open, a section $\xi\in C^{\infty}_{0}(U,L^{2})$
belongs to the submodule $C_{0}^{\infty}(U,\overline{\Sigma{\mathcal{E}}})$ if
and only if each Fourier component ${\mathbb{F}}(\xi)(n)$ as in (5.3) belongs
to the closure in $C_{0}^{\infty}(U,{\mathbb{C}})$ of the submodule generated
by the multiples of the function $\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)$ ($\zeta(z)$ is the Riemann zeta function).

###### Proof.

(i) holds since the Fourier transform is a unitary isomorphism.

(ii) The sheaf ${\mathcal{L}}^{2}$ on $[0,\infty)$ is defined by associating
to an open subset $U\subset[0,\infty)$ the space
${\mathcal{F}}(U)=C_{0}^{\infty}(U,L^{2})$ of smooth sections vanishing at
$L=0$ of the vector bundle $L^{2}$. The action of ${\mathbb{N}}^{\times}$ on
${\mathcal{L}}^{2}$ is given, for $n\in{\mathbb{N}}^{\times}$ and for any pair
of opens $U$ and $U^{\prime}$ of $[0,\infty)$, with $nU\subset U^{\prime}$, by

(5.5) ${\mathcal{F}}(U,n):C_{0}^{\infty}(U^{\prime},L^{2})\to
C_{0}^{\infty}(U,L^{2}),\qquad{\mathcal{F}}(U,n)(\xi)(x)=\sigma_{n}(\xi(nx)).$

Note that with $\mu=\exp x$ one has $\xi(nx)\in L^{2}(C_{\mu^{n}})$ and
$\sigma_{n}(\xi(nx))\in L^{2}(C_{\mu})$. By construction one has:
$\sigma_{n}\sigma_{m}=\sigma_{nm}$, thus the above action of
${\mathbb{N}}^{\times}$ turns ${\mathcal{L}}^{2}$ into a sheaf on
${\mathscr{S}}={[0,\infty)\rtimes{{\mathbb{N}}^{\times}}}$.
(iii) By Lemma 5.1 (i), ${\mathcal{E}}(f)(u)$ is pointwise well-defined, it is
$O(u^{1/2})$ for $u\to 0$, and of rapid decay for $u\to\infty$. By (ii) of the
same lemma one has

${\mathbb{F}}(\Sigma_{\mu}({\mathcal{E}}(f)))(n)=L^{-\frac{1}{2}}\int_{C_{\mu}}\Sigma_{\mu}({\mathcal{E}}(f))(u)u^{-\frac{2\pi
in}{L}}d^{*}u=L^{-\frac{1}{2}}\int_{{\mathbb{R}}_{+}^{*}}{\mathcal{E}}(f)(u)u^{-\frac{2\pi
in}{L}}d^{*}u.$

It then follows from [7] (see $(6.4)$ which is valid for $z=\frac{2\pi
n}{L}\in{\mathbb{R}}$) that

(5.6)
${\mathbb{F}}(\Sigma_{\mu}({\mathcal{E}}(f)))(n)=L^{-\frac{1}{2}}\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)\int_{{\mathbb{R}}_{+}^{*}}u^{\frac{1}{2}}f(u)u^{-\frac{2\pi
in}{L}}d^{*}u.$

Since $f\in{{\mathcal{S}}^{\rm ev}_{0}}$, with $w(f)(u):=u^{1/2}f(u)$, the
multiplicative Fourier transform ${\mathbb{F}}(w(f))=\psi$,
$\psi(z):=\int_{{\mathbb{R}}_{+}^{*}}f(u)u^{\frac{1}{2}-iz}d^{*}u$ is
holomorphic in the complex half-plane defined by $\Im(z)>-5/2$ [7]. Moreover,
by construction ${{\mathcal{S}}^{\rm ev}_{0}}$ is stable under the operation
$f\mapsto u\partial_{u}f+\frac{1}{2}f$, hence $w({{\mathcal{S}}^{\rm
ev}_{0}})$ is stable under $f\mapsto u\partial_{u}f$. This operation
multiplies ${\mathbb{F}}(w(f))(z)=\psi(z)$ by $iz$. This argument shows that
for any integer $m>0$, $z^{m}\psi(z)$ is bounded in a strip around the real
axis and hence that the derivative $\psi^{(k)}(s)$ is $O(|s|^{-m})$ on
${\mathbb{R}}$, for any $k\geq 0$. By applying classical estimates due to
Lindelof [11], (see [1] inequality (56)), the derivatives
$\zeta^{(m)}(\frac{1}{2}+iz)$ are $O(|z|^{\alpha})$ for any $\alpha>1/4$. Thus
all derivatives $\partial_{L}^{m}$ of the function (5.6), now re-written as
$h(L,n):=L^{-\frac{1}{2}}\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)\psi(\frac{2\pi n}{L})$, are sequences of rapid decay as
functions of $n\in{\mathbb{Z}}$. It follows that $\Sigma{\mathcal{E}}(f)$ is a
smooth (global) section of the vector bundle $L^{2}$ over $(0,\infty)$.
Moreover, when $n\neq 0$ the function $h(L,n)$ tends to $0$ when $L\to 0$ and
the same holds for all derivatives $\partial_{L}^{m}h(L,n)$. In fact, for any
$m,k\geq 0$, one has

$\sum_{n\neq 0}|\partial_{L}^{m}h(L,n)|^{2}=O(L^{k})\ \ \rm{when}\ \ L\to 0.$

This result is a consequence of the rapid decay at $\infty$ of the derivatives
of the function $\psi$, and the above estimate of $\zeta(z)$ and its
derivatives. For $n=0$ one has
$h(L,0)=L^{-\frac{1}{2}}\zeta(\frac{1}{2})\psi(0)$.
(iv) For any open subset $U\subset[0,\infty)$ the vector space
$C_{0}^{\infty}(U,L^{2})$ admits a natural Frechet topology with generating
seminorms of the form ($K\subset U$ compact subset)

(5.7) $p_{K}^{(n,m)}(\xi):=\max_{K}L^{-n}\|\partial_{L}^{m}\xi(L)\|_{L^{2}}.$

One obtains a space of smooth sections
$C_{0}^{\infty}(U,\Sigma{\mathcal{E}})\subset C_{0}^{\infty}(U,L^{2})$ defined
as sums of products $\sum h_{j}\Sigma{\mathcal{E}}(f_{j})$, with
$f_{j}\in{{\mathcal{S}}^{\rm ev}_{0}}$ and $h_{j}\in C_{0}^{\infty}(U,L^{2})$.
The map $\sigma_{n}:L^{2}(C_{\mu^{n}})\to L^{2}(C_{\mu})$ in (ii) is
continuous, and from the equality
$\sigma_{n}\circ\Sigma_{\mu^{n}}=\Sigma_{\mu}$ it follows (here we use the
notations as in the proof of (ii)) that the sections $\xi\in
C^{\infty}(U^{\prime},L^{2}(C^{\prime}))$ which belong to
$C^{\infty}(U^{\prime},\overline{\Sigma{\mathcal{E}}({{\mathcal{S}}^{\rm
ev}_{0}})})$ are mapped by ${\mathcal{F}}(U,n)$ inside
$C^{\infty}(U,\overline{\Sigma{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}})})$.
In this way one obtains a sheaf
$\overline{\Sigma{\mathcal{E}}}\subset{\mathcal{L}}^{2}$ of
${\mathcal{O}}$-modules over ${\mathscr{S}}$.
(v) Let $\xi\in H^{0}(U,\overline{\Sigma{\mathcal{E}}})$. By hypothesis, $\xi$
is in the closure of $C_{0}^{\infty}(U,\Sigma{\mathcal{E}}({{\mathcal{S}}^{\rm
ev}_{0}}))\subset C_{0}^{\infty}(U,L^{2})$ for the Frechet topology. The
Fourier components of $\xi$ define continuous maps in the Frechet topology,
thus it follows from (5.6) that the functions $f_{n}={\mathbb{F}}(\xi)(n)$ are
in the closure, for the Frechet topology on $C_{0}^{\infty}(U,{\mathbb{C}})$,
of $C_{0}^{\infty}(U,{\mathbb{C}})g_{n}$, where
$g_{n}(L):=\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)$ is a multiplier of
$C_{0}^{\infty}(U,{\mathbb{C}})$. This conclusion holds thanks to the moderate
growth of the Riemann zeta function and its derivatives on the critical line.
Conversely, let $\xi\in C_{0}^{\infty}(U,L^{2})$ be such that each of its
Fourier components ${\mathbb{F}}(\xi)(n)$ belongs to the closure for the
Frechet topology of $C_{0}^{\infty}(U,{\mathbb{C}})$, of
$C_{0}^{\infty}(U,{\mathbb{C}})g_{n}$. Let $\rho\in
C_{c}^{\infty}([0,\infty),[0,1])$ defined to be identically equal to $1$ on
$[0,1]$ and with support inside $[0,2]$. The functions
$\alpha_{k}(x):=\rho((kx)^{-1})$ ($k>1$) fulfill the following three
properties

  1. (1)

$\alpha_{k}(x)=0,\quad\forall x<(2k)^{-1}$, $\alpha_{k}(x)=1,\quad\forall
x>k^{-1}$.

  2. (2)

$\alpha_{k}\in C_{0}^{\infty}([0,\infty))$.

  3. (3)

For all $m>0$ there exists $C_{m}<\infty$ such that
$|x^{2m}\partial_{x}^{m}\alpha_{k}(x)|\leq C_{m}k^{-1}$ $\forall
x\in[0,1],k>1$.

To justify (3), note that
$x^{2}\partial_{x}f((kx)^{-1})=-k^{-1}f^{\prime}((kx)^{-1})$ and that the
derivatives of $\rho$ are bounded. Thus one has
$|(x^{2}\partial_{x})^{m}\alpha_{k}(x)|\leq\|\rho^{(m)}\|_{\infty}k^{-m}$
$\forall x\in[0,\infty),~{}k>1$ which implies (3) by induction on $m$.
Thus, when $k\to\infty$ one has $\alpha_{k}\xi\to\xi$ in the Frechet topology
of $C_{0}^{\infty}(U,L^{2})$. This is clear if $0\notin U$ since then, on any
compact subset $K\subset U$, all $\alpha_{k}$ are identically equal to $1$ for
$k>(\min K)^{-1}$. Assume now that $0\in U$ and let $K=[0,\epsilon]\subset U$.
With the notation of (5.7) let us show that
$p_{K}^{(n,m)}((\alpha_{k}-1)\xi)\to 0$ when $k\to\infty$. Since
$\alpha_{k}(x)=1,\,\forall x>k^{-1}$ one has, using the finiteness of
$p_{K}^{(n+1,m)}(\xi)$

$\max_{K}L^{-n}\|((\alpha_{k}-1)\partial_{L}^{m}\xi)(L)\|_{L^{2}}\leq\max_{(0,1/k]}L^{-n}\|(\partial_{L}^{m}\xi)(L)\|_{L^{2}}\stackrel{{\scriptstyle
k\to\infty}}{{\to}}0.$

Then one obtains

$L^{-n}\partial_{L}^{m}((\alpha_{k}-1)\xi)(L)=L^{-n}((\alpha_{k}-1)\partial_{L}^{m}\xi)(L)+\sum_{1}^{m}L^{-n}{m\choose
j}(\partial_{L}^{j}\alpha_{k})(L)(\partial_{L}^{m-j}\xi)(L).$

Thus using (3) above and the finiteness of the norms $p_{K}^{(n+2j,m-j)}(\xi)$
one derives: $p_{K}^{(n,m)}((\alpha_{k}-1)\xi)\to 0$ when $k\to\infty$. It
remains to show that $\alpha_{k}\xi$ belongs to the submodule
$C_{0}^{\infty}(U,\overline{\Sigma{\mathcal{E}}})$. It is enough to show that
for $K\subset(0,\infty)$ a compact subset with $\min K>0$, one can approximate
$\xi$ by elements of $C_{0}^{\infty}(U,\Sigma{\mathcal{E}})$ for the norm
$p_{K}^{(0,m)}$. Let $P_{N}$ be the orthogonal projection in $L^{2}(C_{\mu})$
on the finite-dimensional subspace determined by the vanishing of all Fourier
components ${\mathbb{F}}(\xi)(\ell)$ for any $\ell,|\ell|>N$. Given $L\in K$
and $\epsilon>0$ there exists $N(L,\epsilon)<\infty$ such that

(5.8) $\|(1-P_{N})\partial_{L}^{j}\xi(L)\|<\epsilon\quad\forall j\leq m,\
N\geq N(L,\epsilon).$

The smoothness of $\xi$ implies that there exists an open neighborhood
$V(L,\epsilon)$ of $L$ such that (5.8) holds in $V(L,\epsilon)$. The
compactness of $K$ then shows that there exists a finite $N_{K}$ such that

$\|(1-P_{N})\partial_{L}^{j}\xi(L)\|<\epsilon\quad\forall j\leq m,\ L\in K,\
N\geq N_{K}.$

It now suffices to show that one can approximate $P_{N}\xi$, for the norm
$p_{K}^{(0,m)}$, by elements of $C_{0}^{\infty}(U,\Sigma{\mathcal{E}})$. To
achieve this result, we let $L_{0}\in K$ and $\delta_{j}\in
C_{c}^{\infty}({\mathbb{R}}_{+}^{*})$, $|j|\leq N$ be such that

${\mathbb{F}}(\delta_{j})\left(\frac{2\pi
j^{\prime}}{L_{0}}\right)=\begin{cases}1&j=j^{\prime}\\\ 0&j\neq
j^{\prime}\end{cases}\quad\forall j^{\prime},|j^{\prime}|\leq N,\ \
\int_{{\mathbb{R}}_{+}^{*}}u^{1/2}\delta_{j}(u)d^{*}u=0\quad\forall j,|j|\leq
N.$

One construct $\delta_{j}$ starting with a function $h\in
C_{c}^{\infty}({\mathbb{R}}_{+}^{*})$ such that
${\mathbb{F}}(h)\left(\frac{2\pi j}{L_{0}}\right)\neq 0$ and acting on $h$ by
a differential polynomial whose effect is to multiply ${\mathbb{F}}(h)$ by a
polynomial vanishing on all $\frac{2\pi j^{\prime}}{L_{0}}$, $j^{\prime}\neq
j$ and at $i/2$. By hypothesis each Fourier component ${\mathbb{F}}(\xi)(n)$
belongs to the closure in $C_{0}^{\infty}(U,{\mathbb{C}})$ of the multiples of
the function $\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)$. Thus, given
$\epsilon>0$ one has functions $f_{n}\in C_{0}^{\infty}(U,{\mathbb{C}})$,
$|n|\leq N$ such that

$\max_{K}|\partial_{L}^{j}\left({\mathbb{F}}(\xi)(n)-\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)f_{n}(L)\right)|\leq\epsilon\quad\forall j\leq m,\quad|n|\leq N.$

We now can find a small open neighborhood $V$ of $L_{0}$ and functions
$\phi_{j}\in C^{\infty}(V)$, $|j|\leq N$ such that

(5.9) $\sum\phi_{j}(L){\mathbb{F}}(\delta_{j})\left(\frac{2\pi
n}{L}\right)=L^{\frac{1}{2}}f_{n}(L)\quad\forall L\in V.$

This is possible because the determinant of the matrix
$M_{n,j}(L)={\mathbb{F}}(\delta_{j})\left(\frac{2\pi n}{L}\right)$ is non-zero
in a neighborhood of $L_{0}$ where $M_{n,j}(L_{0})$ is the identity matrix.
The even functions $d_{j}(u)$ on ${\mathbb{R}}$, which agree with
$u^{-1/2}\delta_{j}(u)$ for $u>0$, are all in ${{\mathcal{S}}^{\rm ev}_{0}}$
since
$\int_{\mathbb{R}}d_{j}(x)dx=2\int_{{\mathbb{R}}_{+}^{*}}u^{1/2}\delta_{j}(u)d^{*}u=0$.
One then has

${\mathbb{F}}(\Sigma_{\mu}({\mathcal{E}}(d_{j})))(n)=L^{-\frac{1}{2}}\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right){\mathbb{F}}_{\mu}(\delta_{j})\left(\frac{2\pi n}{L}\right)$

by (5.6), and by (5.9) one gets

$\sum\phi_{j}(L){\mathbb{F}}(\Sigma_{\mu}({\mathcal{E}}(d_{j})))(n)=\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)f_{n}(L)\quad\forall L\in V.$

One finally covers $K$ by finitely many such open sets $V$ and use a partition
of unity subordinated to this covering to obtain smooth functions
$\varphi_{\ell}\in C_{c}^{\infty}(0,\infty)$, $g_{\ell}\in{{\mathcal{S}}^{\rm
ev}_{0}}$ such that the Fourier component of index $n$, $|n|\leq N$, of
$\sum\varphi_{\ell}\Sigma{\mathcal{E}}(g_{\ell})$ is equal to
$\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)f_{n}(L)$ on $K$. This shows
that $\xi$ belongs to the closure of
$C_{0}^{\infty}(U,\Sigma{\mathcal{E}}({{\mathcal{S}}^{\rm ev}_{0}}))\subset
C_{0}^{\infty}(U,L^{2})$. ∎

We recall that the space of global sections
$H^{0}({\mathcal{T}},{\mathcal{F}})$ of a sheaf of sets ${\mathcal{F}}$ in a
Grothendieck topos ${\mathcal{T}}$ is defined to be the set ${\rm
Hom}_{{\mathcal{T}}}(1,{\mathcal{F}})$, where $1$ denotes the terminal object
of ${\mathcal{T}}$. For ${\mathcal{T}}={\mathscr{S}}$ and ${\mathcal{F}}$ a
sheaf of sets on $[0,\infty)$, $1$ assigns to an open set $U\subset[0,\infty)$
the single element $*$, on which ${\mathbb{N}}^{\times}$ acts as the identity.
Thus, we understand an element of ${\rm Hom}_{{\mathscr{S}}}(1,{\mathcal{F}})$
as a global section $\xi$ of ${\mathcal{F}}$, where ${\mathcal{F}}$ is viewed
as a sheaf on $[0,\infty)$ invariant under the action of
${\mathbb{N}}^{\times}$.

With the notations of Proposition 5.2 and for $\xi\in{\rm
Hom}_{{\mathscr{S}}}(1,{\mathcal{L}}^{2})$, we write
$\widehat{\xi}(L,n):={\mathbb{F}}(\xi)(n)$ for the (multiplicative) Fourier
components of $\xi$. Then we have

######  Lemma 5.3.

The following facts hold

  1. (i)

The map

(5.10) $\gamma:H^{0}({\mathscr{S}},{\mathcal{L}}^{2})\to
C_{0}^{\infty}([0,\infty),{\mathbb{C}})\times
C_{0}^{\infty}([0,\infty),{\mathbb{C}})\qquad\gamma(\xi)=\widehat{\xi}(\pm 1)$

is an isomorphism of ${\mathbb{C}}$-vector spaces.

  2. (ii)

The subspace $\gamma(H^{0}({\mathscr{S}},\overline{\Sigma{\mathcal{E}}}))$ is
the closed ideal (for the Frechet topology on
$C_{0}^{\infty}([0,\infty),{\mathbb{C}})$), generated by the multiplication
with the functions $\zeta\left(\frac{1}{2}\mp\frac{2\pi i}{L}\right)$.

###### Proof.

(i) Let $\xi\in{\rm Hom}_{{\mathscr{S}}}(1,{\mathcal{L}}^{2})$: this is a
global section $\xi\in C_{0}^{\infty}([0,\infty),L^{2})$ invariant under the
action of ${\mathbb{N}}^{\times}$, i.e. such that $\sigma_{n}(\xi(nL))=\xi(L)$
for all pairs $(L,n)$. The Fourier components $\widehat{\xi}(L,n)$ of any such
section are smooth functions of $L\in[0,\infty)$ vanishing at $L=0$, for
$n\neq 0$, as well as all their derivatives. The equality
$\sigma_{n}(\xi(L))=\xi(L/n)$ entails, for $n>0$,

$\displaystyle\widehat{\xi}(L,n)$
$\displaystyle=L^{-\frac{1}{2}}\int_{C_{\mu}}\xi(L)(u)u^{-\frac{2\pi
in}{L}}d^{*}u=L^{-\frac{1}{2}}\int_{C_{\mu}^{1/n}}\xi(L/n)(u)u^{-\frac{2\pi
i}{L}}d^{*}u=$ $\displaystyle=n^{-\frac{1}{2}}\widehat{\xi}(L/n,1).$

This shows that the $\widehat{\xi}(L,n)$ are uniquely determined, for $n>0$ by
the function $\widehat{\xi}(L,1)$ and, for $n<0$, by the function
$\widehat{\xi}(L,-1)$. With $g(L)=\widehat{\xi}(L,0)$ one has:
$g(L)=n^{-\frac{1}{2}}g(L/n)$ for all $n>0$. This implies, since
${\mathbb{Q}}_{+}^{*}$ is dense in ${\mathbb{R}}_{+}^{*}$ and $g$ is assumed
to be smooth, that $g$ is proportional to $L^{-\frac{1}{2}}$ and hence
identically $0$, since it corresponds to a global section smooth at
$0\in[0,\infty)$. This argument proves that $\gamma$ is injective. Let us show
that $\gamma$ is also surjective. Given a pair of functions $f_{\pm}\in
C_{0}^{\infty}([0,\infty),{\mathbb{C}})$ we construct a global section $\xi\in
H^{0}({\mathscr{S}},{\mathcal{L}}^{2})$ such that $\gamma(\xi)=(f_{+},f_{-})$.
One defines $\xi(L)\in L^{2}(C_{\mu})$ by by means of its Fourier components
set to be $\widehat{\xi}(L,0):=0$, and for $n\neq 0$ by

$\widehat{\xi}(L,n):=|n|^{-\frac{1}{2}}f_{\rm sign(n)}(L/n).$

Since $f_{\pm}(x)$ are of rapid decay for $x\to 0$,
$\sum|\widehat{\xi}(L,n)|^{2}<\infty$, thus $\xi(L)\in L^{2}(C_{\mu})$. All
derivatives of $f_{\pm}(x)$ are also of rapid decay for $x\to 0$, thus all
derivatives $\partial_{L}^{k}(\xi(L))$ belong to $L^{2}(C_{\mu})$ and that the
$L^{2}$-norms $\|\partial_{L}^{k}(\xi(L))\|$ are of rapid decay for $L\to 0$.
By construction $\sigma_{n}(\xi(L))=\xi(L/n)$, which entails $\xi\in
H^{0}({\mathscr{S}},{\mathcal{L}}^{2})$ with $\gamma(\xi)=(f_{+},f_{-})$.
(ii) Let $\xi\in H^{0}({\mathscr{S}},\overline{\Sigma{\mathcal{E}}})$. By
Proposition 5.2 (v), the functions $f_{\pm}=\widehat{\xi}(L,\pm 1)$ are in the
closure, for the Frechet topology on
$C_{0}^{\infty}([0,\infty),{\mathbb{C}})$, of the ideal generated by the
functions $\zeta\left(\frac{1}{2}\mp\frac{2\pi i}{L}\right)$. Conversely, let
$\xi\in H^{0}({\mathscr{S}},{\mathcal{L}}^{2})$ and assume that $\gamma(\xi)$
is in the closed submodule generated by multiplication with
$\zeta\left(\frac{1}{2}\mp\frac{2\pi i}{L}\right)$. The
${\mathbb{N}}^{\times}$-invariance of $\xi$ implies
$\widehat{\xi}(L,n)=|n|^{-\frac{1}{2}}\widehat{\xi}(L/|n|,{\rm sign}(n))$ for
$n\neq 0$. Thus the Fourier components $\widehat{\xi}(L,n)$ belong to the
closure in $C_{0}^{\infty}(U,{\mathbb{C}})$ of the multiples of the function
$\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)$, then Proposition 5.2 (v)
again implies $\xi\in H^{0}({\mathscr{S}},\overline{\Sigma{\mathcal{E}}})$.∎

The action of ${\mathbb{R}}_{+}^{*}$ on the sheaf ${\mathcal{L}}^{2}$ is given
by the action $\vartheta$ on the Fourier components of its sections $\xi$.
With $\mu=\exp L$, $L\in(0,\infty)$, $n\in{\mathbb{N}}^{*}$ and
$\lambda\in{\mathbb{R}}^{*}_{+}$, this is

(5.11) $\displaystyle\widehat{\vartheta(\lambda)\xi}(L,n)$
$\displaystyle=L^{-\frac{1}{2}}\int_{C_{\mu}}\xi(\lambda^{-1}u)u^{-\frac{2\pi
in}{L}}d^{*}u=\lambda^{-\frac{2\pi
in}{L}}L^{-\frac{1}{2}}\int_{C_{\mu}}\xi(v)v^{-\frac{2\pi in}{L}}d^{*}v=$
$\displaystyle=\lambda^{-\frac{2\pi in}{L}}\widehat{\xi}(L,n)$

The following result explains in particular how the quotient sheaf
${\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}$ on ${\mathscr{S}}$ handles
eventual multiplicities of critical zeros of the zeta function.

######  Theorem 5.4.

The induced action of ${\mathbb{R}}_{+}^{*}$ on the global sections
$H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}})$ is
canonically isomorphic to the action of ${\mathbb{R}}_{+}^{*}$, via
multiplication with $\lambda^{is}$, on the quotient of the Schwartz space
${\mathcal{S}}({\mathbb{R}})$ by the closure of the ideal generated by
multiples of $\zeta\left(\frac{1}{2}+is\right)$.

###### Proof.

We first show that the canonical map
$q:H^{0}({\mathscr{S}},{\mathcal{L}}^{2})\to
H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}))$ is
surjective. Let $\xi\in
H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}))$: as a
section of ${\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}$ on $[0,\infty)$,
there exists an open neighborhood $V=[0,\epsilon)$ of $0\in[0,\infty)$ and a
section $\eta\in C_{0}^{\infty}(V,L^{2})$ such that the class of $\eta$ in
$C_{0}^{\infty}(V,L^{2}/\overline{\Sigma{\mathcal{E}}})$ is the restriction of
$\xi$ to $V$. The Fourier components $\widehat{\eta}(L,n)$ are meaningful for
$L\in V$. Since $\xi$ is ${\mathbb{N}}^{\times}$-invariant, for any
$n\in{\mathbb{N}}^{\times}$ the class of ${\mathcal{F}}(V/n,n)(\eta)$, with
${\mathcal{F}}(V/n,n)(\eta)(L):=\sigma_{n}(\eta(nL))$ (see (5.4)) is equal to
the class of the restriction of $\eta$ in
$C_{0}^{\infty}(V/n,L^{2}/\overline{\Sigma{\mathcal{E}}})$. We thus obtain

$\eta(L)-{\mathcal{F}}(V/n,n)(\eta)\in
C_{0}^{\infty}(V/n,\overline{\Sigma{\mathcal{E}}})$

Furthermore, the Fourier components of $\alpha={\mathcal{F}}(V/n,n)(\eta)$ are
given by

$\widehat{\alpha}(L,k)=n^{\frac{1}{2}}\widehat{\eta}(nL,nk).$

Thus, the functions

$\widehat{\eta}(L,k)-\widehat{\alpha}(L,k)=\widehat{\eta}(L,k)-n^{\frac{1}{2}}\widehat{\eta}(nL,nk)$

are the Fourier components of an element in
$C_{0}^{\infty}(V/n,\overline{\Sigma{\mathcal{E}}})$. By Proposition 5.2 (v),
these components are in the closure in $C_{0}^{\infty}(V/n,{\mathbb{C}})$ of
the multiples of the function $\zeta\left(\frac{1}{2}-\frac{2\pi
ik}{L}\right)$. For $k=1$ the function
$\widehat{\eta}(L,1)-n^{\frac{1}{2}}\widehat{\eta}(nL,n)$ is in the closure in
$C_{0}^{\infty}(V/n,{\mathbb{C}})$ of the multiples of the function
$\zeta\left(\frac{1}{2}-\frac{2\pi i}{L}\right)$. This implies that
$\widehat{\eta}(L,n)-n^{-\frac{1}{2}}\widehat{\eta}(L/n,1)$ is in the closure
in $C_{0}^{\infty}(V,{\mathbb{C}})$ of the multiples of the function
$\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)$. For $k=-1$ one obtains a
similar result for the Fourier components $\widehat{\eta}(L,n)$, $n<0$. Thus,
again by Proposition 5.2 (v), one knows that the class of $\eta$ in
$C_{0}^{\infty}(V,L^{2}/\overline{\Sigma{\mathcal{E}}}))$ s not altered if one
replaces $\eta$ with $\eta_{1}\in C_{0}^{\infty}(V,L^{2})$

$\widehat{\eta}_{1}(L,n):=|n|^{-\frac{1}{2}}\eta(L/|n|,{\rm sign}(n)).$

Next step is to extend the functions $\eta(L,\pm 1)\in
C_{0}^{\infty}(V,{\mathbb{C}})$ to $f_{\pm}\in
C_{0}^{\infty}([0,\infty),{\mathbb{C}})$ fulfilling the following property.
For any open set $U\subset[0,\infty)$ and a section $\beta\in
C_{0}^{\infty}(U,L^{2})$, with the class of $\beta$ in
$C_{0}^{\infty}(U,L^{2}/\overline{\Sigma{\mathcal{E}}}))$ being the
restriction of $\xi$ to $U$, the functions $\widehat{\beta}(L,\pm
1)-f_{\pm}(L)$ belong to the closure in $C_{0}^{\infty}(U,{\mathbb{C}})$ of
the multiples of the function $\zeta\left(\frac{1}{2}\mp\frac{2\pi
i}{L}\right)$. To construct $f_{\pm}$ one considers the sheaf
${\mathcal{G}}_{\pm}$ which is the quotient of the sheaf of
$C_{0}^{\infty}([0,\infty),{\mathbb{C}})$ functions by the closure of the
ideal subsheaf generated by the multiples of the function
$\zeta\left(\frac{1}{2}\mp\frac{2\pi i}{L}\right)$. Since the latter is a
module over the sheaf of $C^{\infty}$ functions, it is a fine sheaf, thus a
global section of ${\mathcal{G}}_{\pm}$ can be lifted to a function. By
Proposition 5.2 (v), the Fourier components $\widehat{\xi}_{j}(L,\pm 1)$ of
local sections $\xi_{j}$ of $L^{2}$ representing $\xi$ define a global section
of ${\mathcal{G}}_{\pm}$. The functions $f_{\pm}$ are obtained by lifting
these sections. By appealing to Lemma 5.3, we let $\phi\in
H^{0}({\mathscr{S}},L^{2})$ to be the unique global section such that
$\gamma(\phi)=(f_{+},f_{-})$. Then we show that $q(\phi)=\xi$. We have already
proven that the restrictions to $V=[0,\epsilon)$ are the same. Thus it is
enough to show that given $L_{0}>0$ and a lift $\xi_{0}\in
C_{0}^{\infty}(U,L^{2})$ of $\xi$ in a small open interval $U$ containing
$L_{0}$, the difference $\delta=\phi-\xi_{0}$ is a section of
$\overline{\Sigma{\mathcal{E}}}$. Again by Proposition 5.2 (v), it suffices to
show that the Fourier components $\widehat{\delta}(L,n)$ are in the closure of
the ideal generated by multiples of $\zeta\left(\frac{1}{2}-\frac{2\pi
in}{L}\right)$. The ${\mathbb{N}}^{\times}$-invariance of $\xi$ shows that
${\mathcal{F}}(U/n,n)(\xi_{0})$ (see (5.5)) is a lift of $\xi$ in $U/n$. Thus
by the defining properties of the functions $f_{\pm}$ one has

$\widehat{{\mathcal{F}}(U/n,n)(\xi_{0})}(\pm
1)-f_{\pm}\in\overline{C^{\infty}(U,{\mathbb{C}})\zeta_{\pm}},\qquad\text{for}\quad\zeta_{\pm}(L)=\zeta\left(\frac{1}{2}\mp\frac{2\pi
i}{L}\right).$

With a similar argument and using the invariance of $\phi$ under the action of
${\mathcal{F}}(U/n,n)$, one obtains that $\widehat{\delta}(n)$ is in the
closure of the ideal generated by the multiples of
$\zeta\left(\frac{1}{2}-\frac{2\pi in}{L}\right)$.
We have thus proved that $q:H^{0}({\mathscr{S}},{\mathcal{L}}^{2})\to
H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}))$ is
surjective. One then obtains the exact sequence

(5.12) $0\to H^{0}({\mathscr{S}},\overline{\Sigma{\mathcal{E}}}))\to
H^{0}({\mathscr{S}},{\mathcal{L}}^{2})\to
H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}))\to 0.$

This sequence is equivariant for the action (5.11) of $\vartheta$ of
${\mathbb{R}}_{+}^{*}$ on the bundle $L^{2}$. For $h\in
L^{1}({\mathbb{R}}_{+}^{*},d^{*}u)$ one has

(5.13) $\widehat{(\vartheta(h)\xi)}(L,n)={\mathbb{F}}(h)\left(\frac{2\pi
n}{L}\right)\widehat{\xi}(L,n).$

To obtain the required isomorphism between the two spectral realizations, one
uses the isomorphism (5.10) of Lemma 5.3. Denote for short
$(C_{0}^{\infty})^{2}=C_{0}^{\infty}([0,\infty))\times
C_{0}^{\infty}([0,\infty))$. One maps the Schwartz space
${\mathcal{S}}({\mathbb{R}})$ to $(C_{0}^{\infty})^{2}$ by

$\Phi(f)=(\Phi_{+}(f),\Phi_{-}(f)),\qquad\Phi_{\pm}(f)(L):=f\left(\pm\frac{2\pi}{L}\right).$

$\Phi$ is well defined since all derivatives of $\Phi_{\pm}(f)(L)$ tend to $0$
when $L\to 0$ (any function $f\in{\mathcal{S}}({\mathbb{R}})$ is of rapid
decay as well as all its derivatives). The exact sequence (5.12), together
with Lemma 5.3, then gives an induced isomorphism

$\gamma:H^{0}({\mathscr{S}},{\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}))\simeq(C_{0}^{\infty})^{2}/\left(\overline{C_{0}^{\infty}\zeta_{+}}\times\overline{C_{0}^{\infty}\zeta_{-}}\right).$

In turn, the map $\Phi$ induces a morphism

$\tilde{\Phi}:{\mathcal{S}}({\mathbb{R}})/({\mathcal{S}}({\mathbb{R}})\zeta)\to(C_{0}^{\infty})^{2}/\left(\overline{C_{0}^{\infty}\zeta_{+}}\times\overline{C_{0}^{\infty}\zeta_{-}}\right).$

By (5.13) this morphism is equivariant for the action of
${\mathbb{R}}_{+}^{*}$. The map $\Phi$ is not an isomorphism since elements of
its range have finite limits at $\infty$. However it is injective and its
range contains all elements of $(C_{0}^{\infty})^{2}$ which have compact
support. Since $\zeta_{\pm}(L)=\zeta\left(\frac{1}{2}\mp\frac{2\pi
i}{L}\right)$ tends to a finite non-zero limit when $L\to 0$, $\tilde{\Phi}$
is an isomorphism.∎

######  Remark 5.5.

By a Theorem of Whitney (see [12], Corollary 1.7), the closure of the ideal of
multiples of $\zeta\left(\frac{1}{2}+is\right)$ in
${\mathcal{S}}({\mathbb{R}})$ is the subspace of those
$f\in{\mathcal{S}}({\mathbb{R}})$ which vanish of the same order as $\zeta$ at
every (critical) zero $s\in Z$. Thus if any such zero is a multiple zero of
order $m>1$, one finds that the action of ${\mathbb{R}}_{+}^{*}$ on the global
sections of the quotient sheaf
${\mathcal{L}}^{2}/\overline{\Sigma{\mathcal{E}}}$ admits a non-trivial Jordan
decomposition of the form

$\vartheta(\lambda)\xi=\lambda^{is}(\xi+N(\lambda)\xi),\ \ $

with $N(\lambda)^{m}=0$ and $(1+N(u))(1+N(v))=1+N(uv)$ for all
$u,v\in{\mathbb{R}}_{+}^{*}$.

## References

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# Silicon Substitution in Nanotubes and Graphene via Intermittent Vacancies

Heena Inani  Kimmo Mustonen<EMAIL_ADDRESS>Alexander Markevich
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna,
Austria  Er-Xiong Ding  Aalto University School of Science, Department of
Applied Physics, P.O. Box 15100, FI-00076 Aalto, Finland  Mukesh Tripathi
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna,
Austria  Aqeel Hussain  Aalto University School of Science, Department of
Applied Physics, P.O. Box 15100, FI-00076 Aalto, Finland  Clemens Mangler
Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna,
Austria  Esko I. Kauppinen  Aalto University School of Science, Department of
Applied Physics, P.O. Box 15100, FI-00076 Aalto, Finland  Toma Susi  Jani
Kotakoski<EMAIL_ADDRESS>Faculty of Physics, University of
Vienna, Boltzmanngasse 5, 1090 Vienna, Austria

###### Abstract

The properties of single-walled carbon nanotubes (SWCNTs) and graphene can be
modified by the presence of covalently bound impurities. Although this can be
achieved by introducing chemical additives during synthesis, that often
hinders growth and leads to limited crystallite size and quality. Here,
through the simultaneous formation of vacancies with low-energy argon plasma
and the thermal activation of adatom diffusion by laser irradiation, silicon
impurities are incorporated into the lattice of both materials. After an
exposure of $\sim$1 ion/nm2, we find Si substitution densities of 0.15 nm-2 in
graphene and 0.05 nm-2 in nanotubes, as revealed by atomically resolved
scanning transmission electron microscopy. In good agreement with predictions
of Ar irradiation effects in SWCNTs, we find Si incorporated in both mono- and
divacancies, with $\sim$2/3 being of the first type. Controlled inclusion of
impurities in the quasi-1D and 2D carbon lattices may prove useful for
applications such as gas sensing, and a similar approach might also be used to
substitute other elements with migration barriers lower than that of carbon.

††preprint: AIP/123-QED

Graphene Geim and Novoselov (2010) and single-walled carbon nanotubes Iijima
and Ichihashi (1993) (SWCNTs) are among the most studied materials of the last
two decades. Due to the confinement in either one or two dimensions and the
fact that they consist exclusively of surface atoms, their properties such as
electronic transport and chemical reactivity are highly sensitive to any
structural perturbations. Novoselov  _et al._ (2004); Baierle  _et al._ (2001)
Covalent incorporation of foreign atoms within their lattice has thus been
proposed as a viable route to engineer their properties. Ewels and Glerup
(2005); Terrones  _et al._ (2008); Ayala  _et al._ (2010a)

Atomic-scale observations in graphene have shown nitrogen Bangert  _et al._
(2013) (N), boron Bangert  _et al._ (2013) (B), phosphorus Susi  _et al._
(2017) (P), silicon Ramasse  _et al._ (2013) (Si) and germanium Tripathi  _et
al._ (2018a) (Ge) either as naturally occurring or substitutionally implanted
covalent impurities. In SWCNTs, the presence of N, Keskar  _et al._ (2005);
Susi  _et al._ (2011) B, Ayala  _et al._ (2010b) P Campos-Delgado  _et al._
(2010); Ruiz-Soria  _et al._ (2015) and Si Campos-Delgado  _et al._ (2010) has
been spectroscopically detected in chemically synthesized samples, but no
direct transmission electron microscopy (TEM) evidence of their incorporation
in the tube walls has been shown. Only for N has such conclusive evidence been
presented by Arenal et al. Arenal  _et al._ (2014) Si impurities have
attracted particular recent attention due to the possibility of manipulating
them with the focused electron beam. Susi  _et al._ (2014); Dyck  _et al._
(2017); Tripathi  _et al._ (2018b) As a route complementary to chemical
synthesis, Dyck et al. recently used a 100 kV scanning transmission electron
microscopy (STEM) probe to create vacancies in graphene, which they then
managed to fill with Si by irradiating the surrounding amorphous
contamination. Dyck  _et al._ (2017)

Figure 1:  The experimental system consists of the modified aberration-
corrected Nion UltraSTEM 100 scanning transmission electron microscope in
Vienna Hotz  _et al._ (2016) connected to an external plasma chamber via an
ultra-high vacuum transfer line. Both laser sources operate at 445 nm
wavelength with a power tunable up to 6 W (Lasertack GmbH). Tripathi  _et al._
(2017)

Motivated further by computational predictions of vacancy formation by argon
(Ar) ions by Tolvanen et al., Tolvanen  _et al._ (2007) we demonstrate here
the efficient covalent substitution of Si in both graphene and SWCNTs via an
Ar plasma treatment. To allow a direct comparison, nanotubes were first grown
in a floating catalyst reactor using ethanol and ferrocene as carbon and
catalyst precursors Ding  _et al._ (2017) and deposited on commercially
available graphene on silicon nitride electron microscopy supports from Ted
Pella Inc. Laiho  _et al._ (2017); Mustonen  _et al._ (2018) The deposition
was followed by laser annealing in the Nion UltraSTEM 100 column, Tripathi
_et al._ (2017) exposing clean surfaces for later plasma irradiation
experiments (Figure 1). An overview of the resulting clean tubes interfacing
with graphene is shown in Figure 2a and an atomically resolved closeup in Fig.
2b. The images were acquired at 60 keV electron energy and a beam convergence
semiangle of 30 mrad. The scattered electrons were detected with a medium
angle annular dark field (MAADF) detector at an angular range of 60–200 mrad.

The cleaning was followed by plasma irradiation conducted in a purpose-built
plasma target chamber directly connected to the Nion microscope via an ultra-
high vacuum (UHV) transfer system (Fig. 1). The argon plasma (pressure
5$\times$10-6 mbar) was ignited in a microwave cavity on the side of the
chamber and accelerated to a $\sim$50 eV kinetic energy, exposing the sample
to a dose of $\sim$1 ion/nm2. Chemical cross-linking and the buildup of
amorphous contamination during the experiment was mitigated by concurrently
applying $\sim$100 mW of laser power to the sample, which presumably
simultaneously helped to mobilize the Si impurity atoms. Dyck  _et al._ (2017)
This resulted in the preservation of large clean areas, but also patched the
contamination in small pockets as for example those visible in Fig. 2c.
Although not directly observable, these likely correspond to the positions of
the greatest lattice disorder and therefore the highest chemical reactivity.

Figure 2:  (a) STEM/MAADF overview of laser-cleaned SWCNTs on graphene. (b) An
atomically resolved closeup of the interface. (c) An overview of a plasma-
irradiated sample. (d) A closeup showing the presence of impurity atoms after
plasma irradiation.

Even within the atomically clean areas and regardless of the laser
irradiation, some vacancies especially in graphene remain unoccupied (Fig.
S1). We nevertheless find a large number of covalently bound impurity atoms
with a scattering contrast similar to Si Krivanek  _et al._ (2010) (Fig. 2d).
To confirm their identity, we used electron energy loss spectroscopy (EELS) to
acquire the elemental fingerprints of each atom. Ramasse  _et al._ (2013);
Susi  _et al._ (2017) Since our focus was on impurities bound to SWCNTs, we
studied the tubes suspended over the holes in the graphene support (Figure
3a). The EELS system used here Susi  _et al._ (2017) consists of a Gatan PEELS
666 spectrometer with an energy dispersion of 0.5 eV/px and an Andor iXon 897
electron-multiplying camera. A background-subtracted spectrum recorded from
the atom highlighted in Fig. 3a is given in Fig. 3c, indicating based on the
$L$-edge shape and onset at $\sim$99 eV a 4-coordinated Si impurity. Ramasse
_et al._ (2013); Zhou  _et al._ (2012). The spatial distribution of elements
can be likewise analyzed by recording a spectrum from each probe position and
by mapping the corresponding core losses. Using the Si $L$-edge intensity
($\sim$99–200 eV, see Fig. 3c), the Si distribution was mapped and is visible
in Fig. 3b, confirming the chemical identity of the impurity atoms.

Figure 3:  (a) An overview STEM/MAADF image of SWCNTs in vacuum and
incorporating several covalent Si. (b) The mapped Si $L$-edge intensity in the
99–200 eV energy window (128 px $\times$ 128 px). (c) A spectrum acquired from
the atom highlighted in (a) with spectral features consistent with
4-coordinated Si. Ramasse  _et al._ (2013)

We next turn our attention to the structure and abundance of the Si sites. The
two fully saturated substitutions, 3- and 4-coordinated configurations (Si-C3
and Si-C4, respectively) have been identified in graphene. Ramasse  _et al._
(2013); Zhou  _et al._ (2012) Our atomic resolution observations, including
those in Figure 4, confirm that both configurations are also present in Ar-
irradiated SWCNTs. We observed a total of 61 Si sites in 38 tubes (although
the configuration could only be identified for 51), with $\sim$63% being Si-C3
and $\sim$37% Si-C4. These numbers agree remarkably well with the
computationally projected abundance of single and double vacancies formed upon
ion irradiation at energies similar to ours. Tolvanen  _et al._ (2007) We
likewise observed a large number of Si defects in graphene (Supplementary
Figure S1), but their relative abundances were not determined.

Figure 4:  Examples of atomically resolved STEM/MAADF images of (a) Si-C3 and
(b) Si-C4 impurities in SWCNTs.

Interestingly, regardless of the smaller energy required to displace C atoms
from SWCNTs, Krasheninnikov  _et al._ (2005) this was not reflected in the
relative impurity density. We studied a total SWCNT surface area of 1200 nm2
that contained the aforementioned 61 Si atoms, corresponding to an areal
density of $\sim$0.05 nm-2. A graphene surface area of 1365 nm2 contained more
than three times as many Si impurities, 210 atoms in total, giving a threefold
areal density of $\sim$0.15 nm-2. While Si impurities are commonly found on
graphene Ramasse  _et al._ (2013); Zhou  _et al._ (2012); Susi  _et al._
(2014) (although their origin remains unclear), they are not present on SWCNTs
such as ours that have not undergone liquid dispersion. It is thus likely that
to reach the vacancies created on nanotubes, the Si adatoms are first required
to migrate over the graphene surface.

To study the migration energetics, we ran density functional theory
(DFT)-based atomistic simulations with the projector-augmented wave method
implemented in the GPAW package, Enkovaara  _et al._ (2010) and calculated the
binding energies ($E_{\text{b}}$) and migration barriers ($E_{\text{m}}$) of
Si adatoms on graphene and on a set of achiral SWCNTs (Table 1 and Figure 5).
We used the revPBE Hammer  _et al._ (1999) exchange-correlation functional, a
dzp basis set, Larsen  _et al._ (2009) and a grid spacing of 0.2 Å. The length
of the models for armchair and zigzag SWCNTs were 12.35 and 12.83 Å,
respectively. Graphene was modelled using a 7$\times$7 supercell of 98 atoms.
The Brillouin zone was sampled using 8 k-points in the periodic axial
direction for nanotubes and a 6$\times$6$\times$1 k-point grid for graphene.
Migration barriers were obtained with the climbing-image nudged elastic band
method (cNEB). Henkelman  _et al._ (2000)

The lowest energy configurations for Si adatoms on both graphene and nanotubes
correspond to the bridge site, i.e. above a C-C bond. The calculated binding
energy on graphene was 0.34 eV, slightly lower than the previously reported
values (0.44 eV Xian and Chou (2012) and 0.55 eV Pašti  _et al._ (2018)). This
discrepancy appears to arise from differences in exchange-correlation
functionals, revPBE vs. PBE Perdew  _et al._ (1996), since with the latter
$E_{\text{b}}$ = 0.55 eV was also reproduced in our calculations. On the
surface of nanotubes, the binding energies depend on the tube diameter,
chirality and adsorption site. For smaller diameter tubes the calculated
$E_{\text{b}}$ values are significantly higher than those for graphene, while
converging to the graphene value as the tube diameter increases.
Interestingly, the most reactive C-C bridge never lies on the tubes’ axis or
along their circumference, but is instead diagonal (sites 1 and 2 in Fig. 5).
This is different from C adatoms that have been shown to prefer the
circumferential bridge configuration on armchair tubes. Krasheninnikov  _et
al._ (2004)

The calculated energy barriers (Table 1) show that migration on armchair
SWCNTs prefers the direction of the nanotube axis. For the (7,7) and (15,15)
tubes, migration barriers along the path 1-2 are only 0.08 eV and 0.06 eV,
respectively, and are therefore very close to the value in graphene (0.06 eV
in our calculations, in excellent agreement with previously reported values of
0.06–0.07 eV). Xian and Chou (2012) By contrast, migration paths around the
circumference have barriers 3–5 times higher and are therefore much less
likely to occur. Migration on zigzag tubes is less directed. According to cNEB
calculations for the (12,0) tube, migration along the path 1-2 on the tube
circumference always occurs via configuration 3. Meanwhile, the Si adatom
jumps between positions that are equivalent to sites 1 and 3 can occur in
several directions. The calculated energy barriers for these jumps are higher
than those for migration along the axis of armchair nanotubes and therefore
migration of Si adatoms on zigzag tubes will be slower.

It is important to note that the Si migration barriers are much lower than
those of C adatoms (0.47 eV on graphene Lehtinen  _et al._ (2003) and
$\sim$0.6–1.3 eV on 10–15 Å diameter nanotubes Krasheninnikov  _et al._
(2004)), which may explain why Si is such a common lattice impurity. Ramasse
_et al._ (2013); Susi  _et al._ (2014). Since the barriers in larger diameter
tubes are further nearly as low as in graphene, differences in Si migration
over the surface of the two materials do not provide a direct explanation for
their differing Si concentrations. The fact that we observed many fewer
residual vacancies in SWCNTs, however, suggests that C reconstruction in
nanotubes is in general more efficient. One possible explanation is that
endohedral C adatoms (either endemic or sputtered by the Ar ions) have
migration barriers much smaller than those on the outer wall. Krasheninnikov
_et al._ (2004) Since these atoms are unable to escape from the quasi-one-
dimensional tube, each migrating C will rapidly sample the entire inner
surface and recombine with any vacancy it encounters, Gan  _et al._ (2008)
effectively hindering the incorporation of Si by reducing the number of
available substitution sites.

Figure 5:  Si adsorption sites on (a) (7,7) armchair and (b) (12,0) zigzag SWCNTs reflecting two inequivalent migration paths (1-2 and 1-3). Table 1:  Calculated values of binding energies ($E_{\text{b}}$) and migration barriers ($E_{\text{m}}$) for Si adatoms on graphene and SWNTs. The corresponding adsorption sites are shown in Fig. 5. CNT | Site |  $E_{\text{b}}$, eV | Path |  $E_{\text{m}}$, eV
---|---|---|---|---
Graphene |  | 0.34 |  | 0.06
(7,7) | 1 | 0.82 | 1-2 | 0.08
(7,7) | 3 | 0.66 | 1-3 | 0.43
(7,7) |  |  | 3-1 | 0.27
(15,15) | 1 | 0.46 | 1-2 | 0.06
(15,15) | 3 | 0.32 | 1-3 | 0.26
(15,15) |  |  | 3-1 | 0.12
(12,0) | 1 | 0.95 | 1-3 | 0.35
(12,0) | 3 | 0.74 | 3-1 | 0.13
(26,0) | 1 | 0.43 | 1-3 | 0.19
(26,0) | 3 | 0.33 | 3-1 | 0.09

To summarize, we have used argon ions to create mono- and divacancies in
graphene and single-walled carbon nanotubes and demonstrated the substitution
of Si impurities in both materials in respective concentrations of 0.15 nm-2
and 0.05 nm-2. The captured Si bonded in both 3- and 4-coordinated
configurations, directly identified using atomically resolved scanning
transmission electron microscopy and electron energy loss spectroscopy.
Finally, our atomistic simulations show that Si migration is substantially
faster than that of C. Our findings could enable also other impurities with
similar migration barriers to be captured in the graphitic lattice in greater
quantities than has so far been achieved.

###### Acknowledgements.

M.T. and T.S. acknowledge the Austrian Science Fund (FWF) project P 28322-N36
for funding, and T.S. and A.M. also the European Research Council (ERC) Grant
No. 756277-ATMEN. E.I.K, E.D. and A.H. acknowledge the support from the
Academy of Finland via projects 286546-DEMEC and 292600-SUPER, from TEKES
Finland via projects 3303/31/2015 (CNT-PV) and 1882/31/2016 (FEDOC), and the
Aalto Energy Efficiency (AEF) Research Program through the MOPPI project. A.M.
and T.S. acknowledge the Vienna Scientific Cluster for computer time. J.K.,
H.I. and K.M. were supported by the FWF project I3181 and the Wiener
Wissenschafts-, Forschungs- und Technologiefonds (WWTF) project MA14-009, and
K.M. further by the Finnish Cultural Foundation through a grant from the
Finnish Postdoc Pool and J.K. through FWF project P31605.

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## Supporting Information

Figure S1: STEM/MAADF images of silicon-substituted graphene. a) Overview of
the lattice with many point defects as well as individual and clustered
impurities. b) An individual silicon atom in 4-coordinated configuration. c)
Three nearby silicon impurities.