# Random covers of compact surfaces and smooth linear spectral statistics

By Frédéric Naud  Frédéric Naud
Institut Mathématique de Jussieu
Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05
France<EMAIL_ADDRESS>This paper is dedicated to the memory of
Steve Zelditch, pioneer global analyst who introduced quantum ergodicity into
the realm of pure mathematics.

###### Abstract.

We consider random $n$-covers $X_{n}$ of an arbitrary compact hyperbolic
surface $X$. We show that in the large $n$ regime and small window limit, the
variance of the smooth spectral statistics of the Laplacian $\Delta_{\rho}$
twisted by a unitary abelian character, obey the universal laws of GOE and GUE
random matrices, depending on wether the character $\rho$ preserves or breaks
the time reversal symmetry. We also prove a generalization for higher
dimensional twists valued in compact linear groups. These results confirm a
conjecture of Berry [2, 3] and is a discrete analog of a recent work of
Rudnick [23] for the Weil-Petersson model of random surfaces.

###### Contents

  1. 1 Introduction and main results
  2. 2 Representations, twisted laplace spectrum and trace formula
  3. 3 Character statistics in the large $n$ regime
  4. 4 Bounds on closed geodesics and Chebotarev counting results
    1. 4.1 Some counting bounds
  5. 5 Proof of the main theorem
    1. 5.1 An equidistribution result
    2. 5.2 Expectation and variance in the large $n$ limit
    3. 5.3 The non-primitive contribution
    4. 5.4 Applying the equidistribution Lemma

##  1\. Introduction and main results

It is a long standing conjecture that the fine spectral statistics of
classically chaotic quantum systems should follow, in the semi-classical
regime, a universal behavior related to random matrices. In particular, Berry
gave some theoretical evidence [2, 3] that time reversible systems, in a well
chosen spectral window, should follow the spectral statistics of the Gaussian
orthogonal ensemble (GOE), while non-reversible systems should follow the
statistics of the Gaussian unitary ensemble (GUE). In the mathematical
litterature, there are almost no rigourous results known, except the negative
answer to Berry’s conjecture in the case of arithmetic surfaces by Luo and
Sarnak [13] and the recent preprint of Rudnick [23]. Rudnick’s approach builds
on the Weil-Petersson model of random hyperbolic surfaces and the celebrated
integration formulas of Mirzakhani [17] and most importantly Mirzhakani-Petri
[18]. In this paper, instead of averages over the whole moduli space, we will
perform averages over the finite space of covers of degee $n$ and use the
notion of random covers of compact hyperbolic surfaces defined by Magee, Puder
and the author to study spectral gaps in [15, 14].

Let $\mathbb{H}$ denote the hyperbolic plane endowed with its metric of
constant curvature $-1$, and let $\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$
be a co-compact Fuchsian group so that $X=\Gamma\backslash\mathbb{H}$ is a
compact hyperbolic surface. Let $\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$ be
a group homomorphism, where $\mathcal{S}_{n}$ is the symmetric group of
permutations of $[n]:=\\{1,\ldots,n\\}$. The discrete group $\Gamma$ acts on
$\mathbb{H}\times[n]$ by

$\gamma.(z,j):=(\gamma(z),\phi_{n}(\gamma)(j)).$

The resulting quotient $X_{n}:=\Gamma\backslash\mathbb{H}\times[n]$ is then a
finite cover of degree $n$ of $X$, possibly not connected. By considering the
(finite) space of all homomorphism
$\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$, endowed with the uniform
probability measure, we obtain a notion of Random covering surfaces of degree
$n$, $X_{n}\rightarrow X$. The expectation of any random variable $Y$ with
respect to this uniform measure is denoted by $\mathbb{E}_{n}(Y)$. We point
out that we can also view (up to isometry) the random cover $X_{n}$ as

$X_{n}=\bigsqcup_{k=1}^{p}\Gamma_{k}\backslash\mathbb{H},$

where each $\Gamma_{k}$ is (in general non-normal) subgroup of $\Gamma$ given
by $\Gamma_{k}=\mathrm{Stab}_{\Gamma}(i_{k})=\\{\gamma\in\Gamma\ :\
\phi_{n}(\gamma)(i_{k})=i_{k}\\}$ where $i_{1},\ldots,i_{k}\in[n]$ are
representatives of the orbits of $\Gamma$ (acting on $[n]$ via $\phi_{n}$). In
general, the cover $X_{n}$ is not connected, but it follows directly from
Liebeck and Shalev [12] that the probability that this cover is connected
(i.e. $\Gamma$ acts transitively on $[n]$ via $\phi_{n}$) tends to $1$ as $n$
goes to infinity.

In order to allow us to observe different statistical behaviors, we also fix a
complex unitary representation, globally defined on $\Gamma$,

$\rho:\Gamma\rightarrow GL(V_{\rho}),$

where $V_{\rho}$ is a finite-dimensional complex vector space.

Each surface $X_{n}$ comes with a self-adjoint Laplace operator
$\Delta_{n,\rho}$, the hyperbolic Laplacian on $X_{n}$ twisted by $\rho$. The
spectrum of $\Delta_{n,\rho}$ is denoted (with multiplicity) by

$\left\\{\lambda_{j,n}=\frac{1}{4}+(r_{j,n})^{2},\ j\in\mathbb{N}\right\\},$

where $r_{j,n}\in\mathbb{R}\cup i\mathbb{R}$.

Let $\psi$ be a real valued even smooth test function on $\mathbb{R}$ whose
Fourier transform $\widehat{\psi}$ is compactly supported in $\mathbb{R}$. The
Fourier transform is in our case given by

$\widehat{\psi}(\xi)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\psi(x)e^{-ix\xi}dx.$

Note that $\psi$ is therefore an entire holomorphic function on $\mathbb{C}$.
We take $\alpha,L>0$ and consider the smooth counting function

$N_{n}(L):=\sum_{j=0}^{\infty}\left\\{\psi(L(r_{j,n}-\alpha))+\psi(L(r_{j,n}+\alpha))\right\\}.$

We will fix $\alpha$ (the energy level) and let $L^{-1}$ (the band width) go
to zero and $n$ go to infinity. It is easy to see that for fixed $L$, we have
as $n$ goes to infinity,

$\mathbb{E}_{n}(N_{n}(L))\sim
C_{\alpha}\frac{(g-1)\mathrm{dim}(V_{\rho})n}{L}\int_{\mathbb{R}}\psi(r)dr,$

where $g$ is the genus of the base surface $X$, and
$C_{\alpha}=2\alpha\tanh(\pi\alpha)$. One of the simplest quantity to
investigate is the so-called ”number variance” which measures ”spectral
rigidity”. We will therefore focus on the probabilistic variance

$\mathbb{V}_{n}(L):=\mathbb{E}_{n}\left(\left|N_{n}(L)-\mathbb{E}_{n}(N_{n}(L))\right|^{2}\right).$

Our main result is the following.

######  Theorem 1.1.

Fix $X=\Gamma\backslash\mathbb{H}$ and a unitary abelian character
$\rho:\Gamma\rightarrow\mathbb{C}$ as above. We fix an energy level
$\alpha>0$, then we have the double limit

$\lim_{L\rightarrow\infty}\lim_{n\rightarrow\infty}\mathbb{V}_{n}(L)=\left\\{\Sigma^{2}_{\mathrm{GOE}}(\psi)\
\mathrm{if}\ \rho^{2}=1\ \atop\Sigma^{2}_{\mathrm{GUE}}(\psi)\ \mathrm{if}\
\rho^{2}\neq 1.\right.$

The quantity $\Sigma^{2}_{\mathrm{GOE}}(\psi)$ is the ”smoothed” number
variance of random matrices for the GOE model in the large dimension limit and
given by

$\Sigma^{2}_{\mathrm{GOE}}(\varphi)=2\int_{\mathbb{R}}|x|(\widehat{\psi}(x))^{2}dx,$

while
$\Sigma^{2}_{\mathrm{GUE}}(\psi)=\frac{1}{2}\Sigma^{2}_{\mathrm{GOE}}(\psi)$.

This result is in accordance with Berry’s conjecture: the hypothesis
$\rho^{2}\neq 1$ corresponds to breaking the time reversal symmetry i.e. the
character $\chi$ is sensitive to the orientation of closed geodesics. The
computation of number variances in the model cases of GUE, GOE goes back to
Dyson and Mehta in [7], see Mehta [16], Appendix A.39 and also chapter 16.

What about higher dimensional twists and Laplacians acting on associated flat
vector bundles? We denote by $U(N)$ the unitary group, $SU(N)$ the special
unitary group, $Sp(N)$ the compact symplectic group, $SO(N)$ the special
orthogonal group. Note that $SU(N),SO(N)$ are Lie subgroups of $U(N)$ while
for the compact symplectic group

$Sp(N)=Sp(2N,\mathbb{C})\cap U(2N)$

is a Lie subgroup of $U(2N)$. In the text below $\mathbf{G}$ will denote a
compact connected real Lie subgroup of $U(N)$ for some $N$. Notice that by a
celebrated theorem of Tannaka, see Chevalley [6] , this is automatically a
real algebraic group. We show the following .

######  Theorem 1.2.

Let $\mathbf{G}$ be as above, and assume it is not conjugated to a direct
product of Lie groups. Fix $X=\Gamma\backslash\mathbb{H}$ and a unitary
representation $\rho:\Gamma\rightarrow\mathbf{G}$ whose image is Zariski dense
in $\mathbf{G}$. We fix an energy level $\alpha>0$, then we have the double
limit

$\lim_{L\rightarrow\infty}\lim_{n\rightarrow\infty}\mathbb{V}_{n}(L)=\left\\{\Sigma_{\mathrm{GOE}}^{2}(\psi)\
\mathrm{if}\
g\mapsto\mathrm{Tr}(g)\in\mathbb{R},\atop\Sigma^{2}_{\mathrm{GUE}}(\psi)\
\mathrm{otherwise}.\right.$

In particular, we have the following behavior for classical compact Lie
groups.

  * •

For $\mathbf{G}=U(N)$, or $\mathbf{G}=SU(N)$ with $N\geq 3$, we are in the GUE
case.

  * •

For $\mathbf{G}=SO(N),Sp(N)$, we are in the GOE case.

In a nutshell, if the trace map on $\mathbf{G}$ is real-valued then we get GOE
statistics while if the trace map takes complex values, we have GUE
statistics. Representation of surface groups with Zariski dense images in the
above cases do exist under some mild dimensional assumptions, see for example
in the paper of Kim and Pansu, [10] and also Kishore [11]. More details can be
found at the end of the paper, $\S 5$.

While our results have the same flavor as in [23], here are the main
differences:

  * •

We take finite averages over (increasingly bigger) sets of surfaces. In
particular we cannot rely as much on oscillatory integral techniques and
integration by parts as in the smooth model of Weil-Petersson.

  * •

Our result is at fixed frequency $\alpha$ while Rudnick in [23] lets both
$L=O(\log\alpha)),\alpha$ go to $\infty$ . However, it is not really necessary
at the technical level in [23] to let the energy level $\alpha$ go to infinity
in order to observe the universal behaviour for the variance so both result
are actually the same.

  * •

Our model also allows to study twisted Laplacians and prove the conjectured
effect of time reversal symmetry on the spectral statistics. It would be
interesting to know to what extent Mirzakhani’s formalism needs to be modified
to allow unitary twists in the analysis of Rudnick.

  * •

If $\Gamma$ is an arithmetic groups, then all the random covers $X_{n}$ are
also arithmetic surfaces. It was shown in [13], see also [4], that in the
semi-classical regime, the number variance exhibits some poissonian behaviour,
due to the high multiplicities in lengths of closed geodesics. Our result
shows that in the large $n$ regime we still get GOE/GUE statistics despite
arithmeticity.

The paper is organized as follows. In $\S 2$ we explain how the spectrum of
$\Delta_{n,\rho}$ can be studied via an
$n\times\mathrm{dim}(V_{\rho})$-dimensional unitary twist of the Laplacian on
the base surface $X$, mostly by using the induction formula from
representation theory. We then state the ad-hoc Selberg trace formula that we
will later use in the proof. In $\S 3$ we review the asymptotic statistics (in
the large $n$ regime) of the number of fixed points of ”random” permutation
$\phi_{n}(\gamma)$: when $\gamma$ is primitive, this random variable behaves
essentially like a Poisson variable with parameter $1$. When
$\gamma_{1}\neq\gamma_{2}$ are in two different primitive conjugacy class (and
are not inverses of each other), the associated number of fixed points of
$\phi_{n}(\gamma_{1})$ and $\phi_{n}(\gamma_{2})$ are asymptotically
independent. This will play a key role in the proof of Theorem 1.1. We point
out that a similar behaviour occur in the smooth Weil-Peterssen model where
the core argument in [18] shows independency and Poissonian behaviour for
lengths of simple closed geodesics.

In $\S 4$ we provide some asymptotic results for the (weighted) counting
function for the closed geodesics on $X$, which we derive from some standard
facts on twisted Laplaclans for unitary representations and the trace formula.
Finally, $\S 5$ is devoted to the proof of Theorems 1.1 and 1.2 which in turn
is broken into several steps.

Acknowledgements. The author wants to thank Michael Magee, Stéphane
Nonnenmacher, Doron Puder and Zeev Rudnick for several discussions around this
work.

##  2\. Representations, twisted laplace spectrum and trace formula

Let $\rho:\Gamma\rightarrow U(V_{\rho})$ be a unitary representation of
$\Gamma$, with finite dimensional complex representation space $V_{\rho}$, and
inner hermitian product $\langle.,.\rangle_{\rho}$. The twisted Laplacian
$\Delta_{\rho}$ is the Laplacian acting on smooth functions

$\psi:\mathbb{H}\rightarrow V_{\rho}$

satisfying the equivariance property

$\forall\ \gamma\in\Gamma,\ \psi(\gamma z)=\rho(\gamma)\psi(z).$

We endow this space with an $L^{2}$-norm given by

$\|\psi\|_{\rho}^{2}:=\int_{\mathcal{F}}\|\psi(z)\|_{\rho}^{2}d\mathrm{Vol}(z),$

where $\mathcal{F}\subset\mathbb{H}$ is a compact fundamental domain for
$\Gamma$ and $\mathrm{Vol}$ is hyperbolic volume. This Laplacian
$\Delta_{\rho}$ has a unique self-adjoint extension with discrete spectrum.
Alternatively, one can view $\Delta_{\rho}$ as a self-adjoint Bochner-Laplace
operator associated to a metric connection on the flat vector Bundle
$E_{\rho}\rightarrow X$ associated to the representation $\rho$ of the
fundamental group $\Gamma$. Spectral theory and trace formulas for such
Laplacians are well documented in [8, 26].

We have the following fact.

######  Proposition 2.1.

There exists an $n\times\mathrm{dim}(V_{\rho})$-dimensional complex
representation

$\varrho_{n}:\Gamma\rightarrow U(V_{n})$

such that the spectrum of $\Delta_{n,\rho}$ on the random surface $X_{n}$
coincides (with same multiplicity) with the spectrum of the twisted Laplacian
$\Delta_{\varrho_{n}}$. In addition, we have the character formula for all
$\gamma\in\Gamma$,

$\mathrm{Tr}(\varrho_{n}(\gamma))=\mathrm{Tr}(\rho(\gamma))|\mathrm{Fix}(\phi_{n}(\gamma))|,$

where $\mathrm{Fix}(\phi_{n}(\gamma))$ denotes the set of fixed points in
$[n]$ of the permutation $\phi_{n}(\gamma)$.

Proof. We set $d=\mathrm{dim}(V_{\rho})$. We recall that we have

$X_{n}=\bigsqcup_{k=1}^{p}\Gamma_{k}\backslash\mathbb{H},$

where $\Gamma_{k}=\mathrm{Stab}_{\Gamma}(i_{k})=\\{\gamma\in\Gamma\ :\
\phi_{n}(\gamma)(i_{k})=i_{k}\\}$ where $i_{1},\ldots,i_{k}\in[n]$ are
representatives of the orbits of $\Gamma$ (acting on $[n]$ via $\phi_{n}$).
Working on each connected components of $X_{n}$, it is enough to assume that
$X_{n}$ is actually connected, i.e. $\Gamma$ acts transitively on $[n]$. Let
us set

$\Gamma_{1}:=\mathrm{Stab}_{\Gamma}(1),$

and we write the right cosets decomposition of $\Gamma$ as

$\Gamma=\Gamma_{1}g_{1}\cup\Gamma_{1}g_{2}\cup\ldots\cup\Gamma_{1}g_{n},$

where $g_{1}=Id$ and we have ordered the cosets such that
$\phi_{n}(g_{i})(i)=1$ for all $i=1,\ldots,n$. It is a well known fact (see
for example Venkov [26], page 51) that the Laplacian $\Delta_{\rho}$ on
$X_{n}=\Gamma_{1}\backslash\mathbb{H}$ is unitarily conjugated to the
Laplacian on $E_{\varrho_{n}}$, where $\varrho_{n}$ is the induced
representation of $\rho$ on $\Gamma_{1}$ to $\Gamma$. More precisely,
$\varrho_{n}(\gamma)$ acts on $V_{n}=\mathbb{C}^{d\times n}$ via the block-
matrix

$\mathrm{Mat}_{\mathbb{C}^{d\times
n}}(\varrho_{n}(\gamma))=\left[\widetilde{\rho}(g_{i}\gamma
g_{j}^{-1})\right]_{1\leq i,j\leq n},$

where $\widetilde{\rho}(g)=\rho(g)$ if $g\in\Gamma_{1}$ and
$\widetilde{\rho}(g)=0$ elsewhere. On the diagonal part of this matrix,
observe that if $\phi_{n}(\gamma)(i)\neq i$ then $\widetilde{\rho}(g_{i}\gamma
g_{i}^{-1})=0$ while

$\widetilde{\rho}(g_{i}\gamma g_{i}^{-1})=\rho(g_{i}\gamma
g_{i}^{-1})=\rho(g_{i})\rho(\gamma)\rho(g_{i})^{-1}$

if $\phi_{n}(\gamma)(i)=i$. Computing traces yields

$\mathrm{Tr}(\varrho_{n}(\gamma))=\sum_{i\ :\
\phi_{n}(\gamma)(i)=i}\mathrm{Tr}(\rho(g_{i})\rho(\gamma)\rho(g_{i})^{-1})=\mathrm{Tr}(\rho(\gamma))|\mathrm{Fix}(\phi_{n}(\gamma))|,$

and the proof is done. $\square$

We will use twisted versions of Selberg’s trace formula which is as follows,
see for example in [8, 26]. Let $\varrho:\Gamma\rightarrow U(V_{\varrho})$ be
a general unitary, finite dimensional, representation of $\Gamma$. For all
eigenvalue $\lambda_{j}(\varrho)$ of the Laplacian $\Delta_{\varrho}$, we fix
$r_{j}(\rho)$ such that $\lambda_{j}(\varrho)=1/4+(r_{j}(\varrho))^{2}$. Let
$h$ be an even function whose Fourier transform $\widehat{h}$ is compactly
supported and smooth, then we have the celebrated identity:

######  Theorem 2.2.

(Twisted trace formula)

$\sum_{j=0}^{\infty}h(r_{j}(\varrho))=(g-1)\mathrm{dim}(V_{\varrho})\int_{-\infty}^{\infty}h(r)r\tanh(\pi
r)dr$ $+\sum_{\gamma\in\mathcal{P}}\sum_{k\geq
1}\mathrm{Tr}(\varrho(\gamma^{k}))\frac{\ell(\gamma)\widehat{h}(k\ell(\gamma))}{2\sinh(k\ell(\gamma)/2)}.$

In the above formula, $g$ denotes the genus of the base surface $X$,
$\mathcal{P}$ is the set of primitive conjugacy classes in $\Gamma$ (different
from identity) and if $\gamma\in\mathcal{P}$, $\ell(\gamma)$ denotes the
length of the associated closed geodesic on $X$. The sum over $k$ runs over
all positive integers.

##  3\. Character statistics in the large $n$ regime

In this section we review some key facts on the integer valued random
variables $|\mathrm{Fix}(\phi_{n}(\gamma))|$, for $\gamma\in\Gamma$ and where
$\phi_{n}:\Gamma\rightarrow\mathcal{S}_{n}$ is a random homomorphism. To
simplify further notations, we will denote simply
$|\mathrm{Fix}(\phi_{n}(\gamma))|$ by $F_{n}(\gamma)$. Our analysis will based
on the following facts. We recall that $\mathcal{P}$ denotes the set of
primitive conjugacy classes (different from identity) in $\Gamma$.

######  Proposition 3.1.

Under the above notations, the following holds.

  1. (1)

For all $\gamma\in\mathcal{P}$ and $k\geq 1$, we have as $n$ goes to $\infty$,

$\mathbb{E}_{n}(F_{n}(\gamma^{k}))=d(k)+O\left(\frac{1}{n}\right),$

where $d(k)$ denotes the number of divisors of $k$.

  2. (2)

For all $\gamma_{1},\gamma_{2}\in\mathcal{P}$ such that
$\gamma_{1}\not\in\\{\gamma_{2},\gamma_{2}^{-1}\\}$, for all $k_{1},k_{2}\geq
1$, we have as $n\rightarrow\infty$,

$\mathbb{E}_{n}(F_{n}(\gamma_{1}^{k_{1}})F_{n}(\gamma_{2}^{k_{2}}))=\mathbb{E}_{n}(F_{n}(\gamma_{1}^{k_{1}}))\mathbb{E}_{n}(F_{n}(\gamma_{2}^{k_{2}}))+O\left(\frac{1}{n}\right).$

  3. (3)

For all $k_{1},k_{2}\geq 1$ and $\gamma\in\mathcal{P}$, the limit

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}\left[\left(F_{n}(\gamma^{k_{1}})-\mathbb{E}_{n}(F_{n}(\gamma^{k_{1}}))\right)\left(F_{n}(\gamma^{k_{2}})-\mathbb{E}_{n}(F_{n}(\gamma^{k_{2}}))\right)\right]$
$:=\mathcal{V}(k_{1},k_{2})$

exists. If $k_{1}=k_{2}=1$ then $\mathcal{V}(1,1)=1$. Moreover we have the
formula

$\mathcal{V}(k_{1},k_{2})=\sum_{d|k_{1}\&k_{2}}d.$

Proof. From [22, 15], see for example in [22] Theorem 1.4, we have directly
$(1)$. From [22], Theorem 1.13, we know that if $\gamma_{1},\gamma_{2}$ do not
belong to the same cyclic subgroup, then the random variables
$F_{n}(\gamma_{1}),F_{n}(\gamma_{2})$ are asymptotically independent which
implies $(2)$. Denote by $C_{n,d}(\gamma)$ the number of $d$-cycles of the
random permutation $\phi_{n}(\gamma)$, where $\gamma$ is primitive. As usual
$Z_{\lambda}$ will denote a Poisson variable with parameter $\lambda>0$ whose
law is given by

$\mathbb{P}(Z_{\lambda}=q)=\frac{e^{-\lambda}\lambda^{q}}{q!},$

and satisfies $\mathbb{E}(Z_{\lambda})=\lambda$,
$\mathbb{E}(Z_{\lambda}^{2})=\lambda^{2}+\lambda$. We know from [22, 15],
Theorem 1.14 that as $n$ goes to $\infty$, $C_{n,d}(\gamma)$ converges in
distribution to a Poisson variable $Z_{1/d}$. We point out that this result
was already known for free groups since Nica [19].

Moreover, if $d_{1}\neq d_{2}$ then the variables
$C_{n,d_{1}}(\gamma),C_{n,d_{2}}(\gamma)$ are asymptotically independent.
Using the formula (for $\gamma$ primitive)

$F_{n}(\gamma^{k})=\sum_{d|k}dC_{n,d}(\gamma),$

we get that

$\mathbb{E}_{n}(F_{n}(\gamma^{k_{1}})F_{n}(\gamma^{k_{2}}))=\sum_{d_{1}|k_{1}}\sum_{d_{2}|k_{2}}d_{1}d_{2}\mathbb{E}_{n}(C_{n,d_{1}}(\gamma)C_{n,d_{2}}(\gamma)).$

By the Above facts, if $d_{1}\neq d_{2}$ then

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}(C_{n,d_{1}}(\gamma)C_{n,d_{2}}(\gamma))=\mathbb{E}(Z_{1/d_{1}})\mathbb{E}(Z_{1/d_{2}})=\frac{1}{d_{1}d_{2}}.$

On the other hand, if $d_{1}=d_{2}$ then

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}(C_{n,d_{1}}(\gamma)C_{n,d_{2}}(\gamma))=\mathbb{E}(Z_{1/d_{1}}^{2})=\frac{1}{d_{1}}+\frac{1}{d_{1}^{2}}.$

Using the fact that

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}(F_{n}(\gamma^{k}))=d(k)=\sum_{d|k}1=\sum_{d|k}d\lim_{n\rightarrow\infty}\mathbb{E}_{n}(C_{n,d}(\gamma)),$

we get that

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}\left[\left(F_{n}(\gamma^{k_{1}})-\mathbb{E}_{n}(F_{n}(\gamma^{k_{1}}))\right)\left(F_{n}(\gamma^{k_{2}})-\mathbb{E}_{n}(F_{n}(\gamma^{k_{2}}))\right)\right]$
$=\sum_{d_{1}|k_{1}}\sum_{d_{2}|k_{2}}d_{1}d_{2}\left(\lim_{n\rightarrow+\infty}\mathbb{E}_{n}(C_{n,d_{1}}(\gamma)C_{n,d_{2}}(\gamma))-\frac{1}{d_{1}d_{2}}\right).$
$=\sum_{d|k_{1}\&k_{2}}d,$

and the proof is done. $\square$

##  4\. Bounds on closed geodesics and Chebotarev counting results

We will use throughout this paper the following standard summation by parts
formula: this is standard in any introductory class in analytic number theory.

######  Proposition 4.1.

Let $0<\ell_{0}<\ell_{1}<\ldots<\ell_{n}$ be an increasing sequence of
positive real numbers with $\lim_{n\rightarrow\infty}\ell_{n}=+\infty$, and
let $(c_{n})_{n\in\mathbb{N}}$ be an arbitrary sequence of complex numbers.
Let $f:(0,+\infty)\rightarrow\mathbb{C}$ be a $C^{1}$-function. We have for
all $x\geq 0$,

$\sum_{\ell_{n}\leq
x}c_{n}f(\ell_{n})=A(x)f(x)-\int_{0}^{x}A(t)f^{\prime}(t)dt,$

where

$A(x)=\left\\{\sum_{\ell_{n}\leq x}c_{n}\ \mathrm{if}\ x\geq\ell_{0}\atop 0\
\mathrm{otherwise.}\right.$

It will be sometimes convenient to use the formal notation

$\int_{0}^{x}f(t)dA(t):=\sum_{\ell_{n}\leq x}c_{n}f(\ell_{n}),$

which in the special case when $(c_{n})$ is positive, coincides with an actual
Stieltjes integral.

###  4.1. Some counting bounds

Let $N(T)$ denote the counting function for closed geodesics on
$X=\Gamma\backslash X$ whose length is smaller than $T$ i.e.

$N(T):=\sum_{k\ell(\gamma))\leq T}1,$

where the sum runs over all integers $k\geq 1$ and all $\gamma\in\mathcal{P}$.
One can also consider the primitive counting function

$N^{0}(T):=\sum_{\ell(\gamma))\leq T}1,$

where the sum runs only on primitive conjugacy classes. It has been known
since Huber [9] that as $T\rightarrow+\infty$,

$N^{0}(T)=\frac{e^{T}}{T}(1+o(1)),$

which is commonly referred as ”the prime geodesic theorem”. We point out that
since we have

$N(T)=N^{0}(T)+\sum_{k\geq 2}N^{0}(T/k),$

we actually have as $T\rightarrow+\infty$,

$N(T)=N^{0}(T)+O(Te^{T/2}).$

Later [24, 8] more precise remainders where proved: for all $\epsilon>0$, we
have the following asymptotic formula as $T$ goes to infinity

$N(T)=\mathrm{Li}\left(e^{T}\right)+O\left(e^{\alpha_{0}(\epsilon)T}\right),$
(1)

where $\alpha_{0}(\epsilon)=\max\\{3/4+\epsilon,s_{1}(X)\\}$, where
$0<s_{1}(X)<1$ is related to the first non-trivial eigenvalue of the Laplacian
on $X$ by $s_{1}(1-s_{1})=\lambda_{1}(X)$. From the fact that

$\mathrm{Li}(x):=\int_{2}^{x}\frac{dt}{\log t},$

it is immediate to check that as $T\rightarrow+\infty$,

$N^{0}(T)=\frac{e^{T}}{T}\left(1+O\left(\frac{1}{T}\right)\right),$ (2)

which is sometimes sufficient. Given a unitary representation

$\rho:\Gamma\rightarrow GL(V_{\rho}),$

we set $\chi_{0}(\gamma):=\mathrm{Tr}(\rho(\gamma))$. We then consider the
following counting function

$N_{\chi_{0}}(T):=\sum_{k\ell(\gamma))\leq T}\chi_{0}(\gamma^{k}).$

We will show the following fact.

######  Proposition 4.2.

Assume that $\rho$ does not contain the trivial representation, then there
exists $\beta_{0}=\beta_{0}(\Gamma,\chi_{0})<1$ such that as
$T\rightarrow+\infty$,

$|N_{\chi_{0}}(T)|=O\left(e^{\beta_{0}T}\right).$

Proof. The proof follows from standard facts on the spectrum of the twisted
Laplacian $\Delta_{\rho}$ and will certainly not surprise the experts. We
nevertheless include a concise proof for completeness. Let
$0\leq\lambda_{0}(\rho)\leq\lambda_{1}(\rho)\leq\ldots$ denote the spectrum of
$\Delta_{\rho}$. From the analysis of Phillips-Sarnak [21] in the scalar case
or from Sunada [25] in the general vector case, we know that if $\rho$ does
not contain the trivial representation, then we have $\lambda_{0}(\rho)>0$. To
be more precise, it is actually shown in [25] that if $\mathcal{A}$ is a
finite set of generators of $\Gamma$ then there exists
$C(\mathcal{A},\Gamma)>0$ such that

$C(\mathcal{A},\Gamma)\delta_{\mathcal{A}}(\rho,{\mathbf{1}})^{2}\leq\lambda_{0}(\rho),$

where $\delta_{\mathcal{A}}(\rho,{\mathbf{1}})$ is Khazdan’s distance to the
trivial representation $\bf 1$. Khazdan’s distance is defined by

$\delta_{\mathcal{A}}(\rho,{\mathbf{1}})=\inf_{v\in
V_{\rho}\atop\|v\|=1}\max_{\gamma\in\mathcal{A}}\|\rho(\gamma)v-v\|.$

It is straightworward to check that if
$\delta_{\mathcal{A}}(\rho,{\mathbf{1}})=0$, then one can find $v\neq 0\in
V_{\rho}$ such that for all $\gamma\in\Gamma$, $\rho(\gamma)v=v$ i.e. $\rho$
contains the trivial representation.

The end of the proof now follows from an application of the trace formula.
First observe that we have

$N_{\chi_{0}}(T)=\sum_{k\ell(\gamma))\leq
T}\chi_{0}(\gamma^{k})=\sum_{\gamma\in\mathcal{P}\atop\ell(\gamma)\leq
T}\chi_{0}(\gamma)+O\left(Te^{T/2}\right),$

just by noticing that by the prime orbit theorem we have

$\sum_{k\ell(\gamma))\leq T\atop k\geq 2}1=O\left(Te^{T/2}\right).$

Therefore it is enough to prove the desired result for

$N_{\chi_{0}}^{0}(T):=\sum_{\ell(\gamma))\leq T}\chi_{0}(\gamma).$

Let $\ell_{0}>0$ denote the shortest length of closed geodesics on $X$. Pick
$\ell_{0}<T_{0}<T_{0}+T_{1}$, and let $0\leq q(x)\leq 1$ be an even smooth
compactly supported function on $\mathbb{R}$ such that $q(x)=0$ for all
$x\in[-\ell_{0}/2,+\ell_{0}/2]$ and $q(x)=1$ for all $\ell_{0}\leq|x|\leq
T_{0}$ while being supported inside $\\{|x|\leq T_{0}+T_{1}\\}$. In the latter
we will choose $T_{1}$ as a function of $T_{0}$ satisfying $T_{1}=e^{-\alpha
T_{0}}$, for some $\alpha>0$, while all the bounds will be relevant for
$T_{0}\rightarrow+\infty$. Moreover, it is possible to choose $q$ such that
for all $k$,

$\sup_{\mathbb{R}}|q^{(k)}|\leq C_{k}T_{1}^{-k}.$

Consider now the test function

$\varphi_{T_{0},T_{1}}(x):=\frac{2\cosh(x/2)}{|x|}q(x),$

and set

$h_{T_{0},T_{1}}(u):=\int_{\mathbb{R}}e^{iux}\varphi_{T_{0},T_{1}}(x)dx.$

Applying the trace formula from Theorem 2.2, a direct bound shows that the
finite contribution from the low eigenvalues below $1/4$ satisfies

$\sum_{j\ :\ r_{j}\in
i\mathbb{R}}|h_{T_{0},T_{1}}(r_{j})|=O\left(T_{0}e^{(1/2+s_{0})T_{0}}\right),$

where

$s_{0}=\left\\{\sqrt{1/4-\lambda_{0}}\ \mathrm{if}\ \lambda_{0}\leq 1/4,\atop
0\ \mathrm{otherwise}.\right.$

Note that since $\lambda_{0}>0$, we have $s_{0}<1/2$. Integrating by parts 3
times yields

$h_{T_{0},T_{1}}(u)=O\left(\frac{e^{T_{0}/2}T_{1}^{-3}}{1+|u|^{3}}\right),$

which gives by a crude bound

$\int_{-\infty}^{\infty}h_{T_{0},T_{1}}(r)r\tanh(\pi
r)dr=O\left(e^{T_{0}/2}T_{1}^{-3}\right).$

We recall that by Weyl’s law we have the bound

$\\#\\{j\ :\ |r_{j}|\leq R\\}=O(R^{2}),$

so that a standard summation by parts yields also the crude bound

$\sum_{j\ :\
r_{j}\in\mathbb{R}}|h_{T_{0},T_{1}}(r_{j})|=O\left(e^{T_{0}/2}T_{1}^{-3}\right).$

We now observe that

$\sum_{\gamma\in\mathcal{P}}\sum_{k\geq
1}\chi_{0}(\gamma^{k})\frac{\ell(\gamma)\varphi_{T_{0},T_{1}}(k\ell(\gamma))}{2\sinh(k\ell(\gamma)/2)}=\sum_{\gamma\in\mathcal{P}}\chi_{0}(\gamma^{k})(\tanh(\ell(\gamma)/2))^{-1}q(\ell(\gamma))+O(T_{0}e^{T_{0}/2}).$

By the the fact that

$\frac{1}{\tanh(x)}=1+O(e^{-2x}),$

we have

$\sum_{\gamma\in\mathcal{P}}\chi_{0}(\gamma^{k})(\tanh(\ell(\gamma)/2))^{-1}q(\ell(\gamma))=\sum_{\gamma\in\mathcal{P}}\chi_{0}(\gamma^{k})q(\ell(\gamma))+O\left(\sum_{\gamma}e^{-\ell(\gamma)}q(\ell(\gamma))\right),$

and the latter term is easily shown to be $O(1)$ by summation by parts. By the
very definition of $q$ we have obtained that

$\sum_{\gamma\in\mathcal{P}}\sum_{k\geq
1}\chi_{0}(\gamma^{k})\frac{\ell(\gamma)\varphi_{T_{0},T_{1}}(k\ell(\gamma))}{2\sinh(k\ell(\gamma)/2)}$
$=N^{0}_{\chi_{0}}(T_{0})+O(T_{0}e^{T_{0}/2})+\sum_{T_{0}\leq\ell(\gamma)\leq
T_{0}+T_{1}}\chi_{0}(\gamma^{k})q(\ell(\gamma)).$

By the prime orbit theorem with remainder (1),

$\left|\sum_{T_{0}\leq\ell(\gamma)\leq
T_{0}+T_{1}}\chi_{0}(\gamma^{k})q(\ell(\gamma))\right|\leq
O\left(N^{0}(T_{0}+T_{1})-N^{0}(T_{0})\right)=O(e^{T_{0}}T_{1})+O(e^{\alpha_{0}T_{0}}).$

Gathering all these estimates, we have thus

$N^{0}_{\chi_{0}}(T_{0})=O(T_{0}e^{T_{0}/2})+O(e^{T_{0}}T_{1})+O(e^{\alpha_{0}T_{0}})+O\left(e^{T_{0}/2}T_{1}^{-3}\right)+O\left(T_{0}e^{(1/2+s_{0})T_{0}}\right),$

where $\alpha_{0}<1$, and $s_{0}<1/2$. Setting $\alpha=\frac{1}{8}$ gives

$N^{0}_{\chi_{0}}(T_{0})=O(T_{0}e^{T_{0}/2})+O(e^{\alpha_{0}T_{0}})+O\left(T_{0}e^{(1/2+s_{0})T_{0}}\right)+O\left(e^{\frac{7}{8}T_{0}}\right),$

and the proof is done. $\square$

Notice that we have not attempted at all to optimize the error term since any
exponent $\beta_{0}<1$ will be enough for our application. We will now apply
this result to prove the following asymptotic formula.

######  Proposition 4.3.

(Chebotarev type asymptotic) Assume that $\rho:\Gamma\rightarrow\mathbf{G}$ is
a unitary representation where $\mathbf{G}$ is a compact algebraic Lie
subgroup of $GL_{N}(\mathbb{C})$. Assume that the image of $\Gamma$ by $\rho$
is Zariski dense in $\mathbf{G}$. Let $f(g)=\phi(\mathrm{Tr}(g))$ be a
polynomial function of the trace. Then as $x\rightarrow+\infty$ we have

$\sum_{\ell(\gamma)\leq
x}f(\rho(\gamma))=\left(\int_{\mathbf{G}}f(g)dg\right)\mathrm{Li}(e^{x})+O(e^{\alpha_{0}x}),$

for some $\alpha_{0}<1$ and where $dg$ denotes normalized Haar measure on
$\mathbf{G}$.

Proof. By Peter-Weyl theorem, since $f(g)$ is a smooth central function we
know that we can expand it as

$f(g)=\sum_{\lambda\in\widehat{\mathbf{G}}}c_{\lambda}\mathrm{Tr}(\lambda(g)),$

where $c_{\lambda}\in\mathbb{C}$ the sum runs over all irreducible
representations $\lambda$ of $\mathbf{G}$. This sum is actually finite: indeed
by the polynomial hypothesis we have the finite sum

$f(g)=\sum_{k,l}a_{k,l}(\mathrm{Tr}(g))^{k}(\overline{\mathrm{Tr}(g)})^{l},$

where each $a_{k,l}\in\mathbb{C}$. Let $\mathcal{I}:\mathbf{G}\rightarrow
GL(\mathbb{C}^{N})$ denote the faithfull representation of $\mathbf{G}$ acting
as usual on $\mathbb{C}^{N}$ by left matrix product with column vectors. Then
we have

$(\mathrm{Tr}(g))^{k}(\overline{\mathrm{Tr}(g)})^{l}=\mathrm{Tr}\left(\left(\mathcal{I}^{\otimes
k}\otimes\widehat{\mathcal{I}}^{\otimes l}\right)(g)\right),$

where $\widehat{\mathcal{I}}$ is the contragredient representation of
$\mathcal{I}$, see for example [5] Proposition 2.6. The finite-dimensional
representation

$\mathcal{I}^{\otimes k}\otimes\widehat{\mathcal{I}}^{\otimes l}$

can in turn be written as a finite direct sum of irreducible representations
of $\mathbf{G}$, hence $(\mathrm{Tr}(g))^{k}(\overline{\mathrm{Tr}(g)})^{l}$
can be expressed as a finite linear combination of characters
$\mathrm{Tr}(\lambda(g))$, and so does $f(g)$. The uniqueness of Peter-Weyl
decomposition then implies that only finitely many coefficients $c_{\lambda}$
must be non-vanishing, notice in addition that we have

$c_{\mathrm{trivial}}=\int_{\mathbf{G}}f(g)dg.$

We then write

$\sum_{\ell(\gamma)\leq
x}f(\rho(\gamma))=\sum_{\lambda\in\widehat{\mathbf{G}}}c_{\lambda}\sum_{\ell(\gamma)\leq
x}\mathrm{Tr}(\lambda(\rho(\gamma)))$
$=\left(\int_{\mathbf{G}}f(g)dg\right)\sum_{\ell(\gamma)\leq
x}1+\sum_{\lambda\in\widehat{\mathbf{G}}\atop\lambda\ \mathrm{non\
trivial}}c_{\lambda}\sum_{\ell(\gamma)\leq
x}\mathrm{Tr}(\lambda(\rho(\gamma))).$

Assume now that if $\lambda$ is irreducible and assume that $\lambda\circ\rho$
contains the trivial representation. This means that there exists a non
vanishing vector $v\in V_{\lambda}$ such that for all $\gamma\in\Gamma$,

$\lambda\circ\rho(\gamma)v=v.$

Denote by $\mathcal{Z}_{\lambda}$ the set

$\mathcal{Z}_{\lambda}:=\\{g\in\mathbf{G}\ :\ \lambda(g)v=v\\}.$

Since $\lambda$ is irreducible, $\mathcal{Z}_{\lambda}\subsetneq\mathbf{G}$
and it is a real algebraic subset of $\mathbf{G}$. Indeed all irreducible
representations can be included 111This is a folklore theorem in the
representation theory of compact Lie groups and can be proved in multiple ways
see for example the discussion at
https://mathoverflow.net/questions/58633/does-every-irreducible-
representation-of-a-compact-group-occur-in-tensor-product and references given
here. in

$\mathcal{I}^{\otimes k}\otimes\widehat{\mathcal{I}}^{\otimes l}$

for some $k,l\in\mathbb{N}$, and thus matrix coefficients are Laurent
polynomials in the matrix coefficients of elements of $\mathbf{G}$. This
translates (by taking real and imaginary parts) into real algebraic equations.
Since $\rho(\Gamma)$ is assumed to be Zariski dense, we must have

$\rho(\Gamma)\cap\mathcal{Z}_{\lambda}^{c}\neq\emptyset,$

and thus $\lambda\circ\rho$ cannot contain the trivial representation. The sum
over irreducible $\lambda$ being finite, we can apply the prime orbit theorem
with remainder and Proposition 4.2 to conclude the proof. $\square$

We point out that the leading term in the above asymptotic should follow
directly from a dynamical result of Parry and Pollicott [20], provided that
one can check the weak-mixing property of the associated extension of the
geodesic flow via the Zariski density assumption. We have chosen here a more
self-contained and scenic road which gives a stronger result which might be
useful in future applications.

##  5\. Proof of the main theorem

###  5.1. An equidistribution result

In core of the proof of the main theorem, we will need to use the following
equidistribution result which we state and prove here.

######  Lemma 5.1.

Let $0<\ell_{0}<\ell_{1}<\ldots<\ell_{n}$ be a sequence such that
$\ell_{n}\rightarrow+\infty$ as $n\rightarrow+\infty$ and let $(a_{n})_{n\geq
0}$ be a sequence of positive real numbers. Consider the counting function

$\Pi(x):=\sum_{\ell_{n}\leq x}a_{n},$

and assume that as $x\rightarrow+\infty$, we have
$\Pi(x)\sim\kappa\frac{e^{x}}{x}$, for some $\kappa>0$. Assume that
$g:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ is a continuous function such that
$g(x)=\frac{x^{2}}{e^{x}}\left(1+O(1/x)\right)$ as $x\rightarrow+\infty$. Pick
$\widehat{\psi}$ a positive valued, compactly supported $C^{\infty}$ function
on $\mathbb{R}^{+}$. Fix $\alpha\in\mathbb{R}$. Then as $T\rightarrow+\infty$,
we have

$\sum_{\ell_{n}}a_{n}\cos^{2}(\alpha\ell_{n})g(\ell_{n})\widehat{\psi}\left(\frac{\ell_{n}}{T}\right)\sim\frac{\kappa
T^{2}}{2}\int_{0}^{\infty}u\widehat{\psi}(u)du,$

if $\alpha\neq 0$, while

$\sum_{\ell_{n}}a_{n}\cos^{2}(\alpha\ell_{n})g(\ell_{n})\widehat{\psi}\left(\frac{\ell_{n}}{T}\right)\sim\kappa
T^{2}\int_{0}^{\infty}u\widehat{\psi}(u)du$

if $\alpha=0$.

Proof. First notice that for some constant $C>0$, we have the following bound
valid for all $x\geq 0$,

$\Pi(x)\leq\left\\{C\frac{e^{x}}{x}\ \mathrm{if}\ x\geq 1\atop C\ \mathrm{if}\
0\leq x\leq 1.\right.$

Set

$S_{T}:=\sum_{\ell_{n}}a_{n}\cos^{2}(\alpha\ell_{n})g(\ell_{n})\widehat{\psi}\left(\frac{\ell_{n}}{T}\right).$

By the asymptotic behaviour of $g$, we have as $T\rightarrow\infty$,

$S_{T}=\int_{0}^{\infty}\cos^{2}(\alpha
x)\frac{x^{2}}{e^{x}}\widehat{\psi}\left(\frac{x}{T}\right)d\Pi(x)+O\left(\int_{0}^{C_{1}T}\frac{x}{e^{x}}d\Pi(x)\right),$

for some $C_{1}>0$ which depends on the support of $\widehat{\psi}$. Using the
above bound for $\Pi(x)$ and a summation by parts we get

$\int_{0}^{C_{1}T}\frac{x}{e^{x}}d\Pi(x)=\int_{0}^{C_{1}T}e^{-x}(x-1)\Pi(x)dx=O(T).$

We are left to analyze

$S_{T}^{0}:=\int_{0}^{\infty}\cos^{2}(\alpha
x)\frac{x^{2}}{e^{x}}\widehat{\psi}\left(\frac{x}{T}\right)d\Pi(x).$

Using the identity

$\cos^{2}(z)=\frac{\cos(2z)+1}{2},$

we write

$S_{T}^{0}:=\widetilde{S}_{T}+S_{T}^{1},$

where

$\widetilde{S}_{T}=\frac{1}{2}\int_{0}^{\infty}\frac{x^{2}}{e^{x}}\widehat{\psi}\left(\frac{x}{T}\right)d\Pi(x),\
S_{T}^{1}=\frac{1}{2}\int_{0}^{\infty}\cos(2\alpha
x)\frac{x^{2}}{e^{x}}\widehat{\psi}\left(\frac{x}{T}\right)d\Pi(x).$

We first deal with $\widetilde{S}_{T}$. A summation by parts yields

$\widetilde{S}_{T}=\frac{1}{2}\int_{0}^{\infty}e^{-x}(x^{2}-2x)\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx-\frac{2}{T}\int_{0}^{\infty}\frac{x^{2}}{e^{x}}\widehat{\psi}^{\prime}\left(\frac{x}{T}\right)\Pi(x)dx.$

We then observe that

$\frac{1}{T}\int_{0}^{\infty}\frac{x^{2}}{e^{x}}\widehat{\psi}^{\prime}\left(\frac{x}{T}\right)\Pi(x)dx=O\left(\frac{1}{T}\int_{0}^{\infty}x\left|\widehat{\psi}^{\prime}\left(\frac{x}{T}\right)\right|dx\right)=O(T),$

while

$\int_{0}^{\infty}e^{-x}(x^{2}-2x)\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx=\int_{0}^{\infty}x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx+O(T).$

Therefore as $T\rightarrow+\infty$,

$\widetilde{S}_{T}=\frac{1}{2}\int_{0}^{\infty}x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx+O(T).$

By a change of variable, we have

$\int_{0}^{\infty}x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx=T^{2}\int_{0}^{\infty}f_{T}(u)du,$

where $f_{T}(u)=Tu^{2}e^{-Tu}\widehat{\psi}(u)\Pi(Tu)$. Notice that we have
for fixed $u\geq 0$,

$\lim_{T\rightarrow\infty}f_{T}(u)=\kappa u\widehat{\psi}(u).$

On the other hand we have for all $T$ large, for all $u\geq 0$,

$0\leq f_{T}(u)\leq Cu\widehat{\psi}(u),$

therefore Lebesgue’s dominated convergence theorem yields

$\lim_{T\rightarrow\infty}\frac{\widetilde{S}_{T}}{T^{2}}=\frac{\kappa}{2}\int_{0}^{\infty}u\widehat{\psi}(u)du,$

which is the asymptotic formula expected. If $\alpha=0$, then the proof is
done since $S_{T}^{1}=\widetilde{S}_{T}$. We now deal with $S_{T}^{1}$
assuming $\alpha\neq 0$. A summation by parts combined with the crude bound on
$\Pi(x)$ as above yields for large $T$

$S_{T}^{1}=\frac{1}{2}\int_{0}^{\infty}\cos(2\alpha
x)x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx$
$+\alpha\int_{0}^{\infty}\sin(2\alpha
x)x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx+O(T).$

Let us analyze the term

$I_{T}:=\int_{0}^{\infty}\cos(2\alpha
x)x^{2}e^{-x}\widehat{\psi}\left(\frac{x}{T}\right)\Pi(x)dx,$

the other term involving the sine being similar. By a change of variable we
get

$I_{T}=T^{3}\int_{0}^{\infty}u^{2}e^{-Tu}\cos(2\alpha
Tu)\Pi(Tu)\widehat{\psi}(u)du$ $=T^{2}\int_{0}^{\infty}f_{T}(u)\cos(2\alpha
Tu)du,$

where $f_{T}(u)=Tu^{2}e^{-Tu}\Pi(Tu)\widehat{\psi}(u)$ as above. We then write

$\frac{I_{T}}{T^{2}}=\int_{0}^{\infty}(f_{T}(u)-\kappa
u\widehat{\psi}(u))\cos(2\alpha
Tu)du+\kappa\int_{0}^{\infty}u\widehat{\psi}(u)\cos(2\alpha Tu)du,$

and a single integration by parts shows that

$\int_{0}^{\infty}u\widehat{\psi}(u)\cos(2\alpha
Tu)du=O\left(\frac{1}{T}\right),$

while

$\left|\int_{0}^{\infty}(f_{T}(u)-\kappa u\widehat{\psi}(u))\cos(2\alpha
Tu)du\right|\leq\int_{\mathbb{R}^{+}}|f_{T}(u)-\kappa u\widehat{\psi}(u)|du.$

By applying Lebesgue’s theorem as before, we finally conclude that

$\lim_{T\rightarrow+\infty}\frac{I_{T}}{T^{2}}=0,$

and thus

$\lim_{T\rightarrow+\infty}\frac{S_{T}}{T^{2}}=\lim_{T\rightarrow+\infty}\frac{\widetilde{S}_{T}}{T^{2}}=\frac{\kappa}{2}\int_{0}^{\infty}u\widehat{\psi}(u)du,$

and the proof is done. $\square$

The rest of this section is devoted to the proof of Theroem 1.1 and is
naturally subdivided into three main steps. We recall that we investigate the
spectrum of $\Delta_{n,\rho}$ where $\rho:\Gamma\rightarrow GL(V_{\rho})$ is a
finite dimensional unitary representation. For simplicity we will denote the
character of $\rho$ by

$\chi(\gamma):=\mathrm{Tr}(\rho(\gamma)).$

Let $\psi$ be a $C^{\infty}$ function whose Fourier transform
$\widehat{\psi}=\widehat{\psi}$ is in $C_{0}^{\infty}(\mathbb{R})$, is even,
and takes its values in $\mathbb{R}^{+}$. Notice that $\psi$ is given by

$\psi(r)=\int_{\mathbb{R}}\widehat{\psi}(u)e^{iru}du,$

and is therefore even too. We will apply the (twisted) trace formula to

$h(r)=\psi(L(r-\alpha))+\psi(L(r+\alpha)),$

where $\alpha$ is fixed and $L$ will be taken large. We then have

$\widehat{h}(u)=\frac{2\cos(\alpha
u)}{L}\widehat{\psi}\left(\frac{u}{L}\right),$

so that

$N_{n}(L)=(g-1)n\mathrm{dim}(V_{\rho})\int_{\mathbb{R}}\left(\psi(L(r-\alpha))+\psi(L(r+\alpha))\right)r\tanh(\pi
r)dr+N_{n}^{0}(L),$

where we have set

$N_{n}^{0}(L)=\frac{2}{L}\sum_{\gamma\in\mathcal{P}}\sum_{k\geq
1}\chi(\gamma^{k})F_{n}(\gamma^{k})\frac{\ell(\gamma)\widehat{\psi}(k\ell(\gamma)/L)}{2\sinh(k\ell(\gamma)/2)}\cos(\alpha
k\ell(\gamma)).$

Notice that if $\gamma$ is a representative of a primitive conjugacy class,
then $\gamma^{-1}$ represents another conjugacy class which corresponds to a
closed geodesic with reversed orientation. Therefore the character $\chi$
breaks the time reversal symmetry if and only if it can distinguish the
orientation of some closed geodesics. As emphasized by the physical intuition,
it will prove more convenient to re-write the above sum over non-oriented
primitive closed geodesics: we set
$\mathcal{P}_{0}=\mathcal{P}/\\{\mathrm{Id},\mathrm{Inv}\\}$, where
$\mathrm{Inv}:\mathcal{P}\rightarrow\mathcal{P}$ is the involution given by
the inversion, acting on conjugacy classes. We then have

$N_{n}^{0}(L)=\frac{2}{L}\sum_{\gamma\in\mathcal{P}_{0}}\sum_{k\geq
1}\left(\chi(\gamma^{k})+\overline{\chi(\gamma^{k})}\right)F_{n}(\gamma^{k})\frac{\ell(\gamma)\widehat{\psi}(k\ell(\gamma)/L)}{2\sinh(k\ell(\gamma)/2)}\cos(\alpha
k\ell(\gamma)).$

###  5.2. Expectation and variance in the large $n$ limit

We first observe that by taking expectations and using the fact that for all
$k$,

$\lim_{n\rightarrow}\mathbb{E}_{n}(F_{n}(\gamma^{k}))=d(k)$

exists, we obtain that for large $n$ (here $\alpha,L$ are fixed),

$\mathbb{E}_{n}(N_{n}(L))=(g-1)n\mathrm{dim}(V_{\rho})\int_{\mathbb{R}}\left(\psi(L(r-\alpha))+\psi(L(r+\alpha))\right)r\tanh(\pi
r)dr+O_{L}(1),$

which is a smooth probabilistic Weyl law in the large $n$ regime. The quantum
variance is

$\mathbb{V}_{n}(L):=\mathbb{E}_{n}\left(\left|N_{n}(L)-\mathbb{E}_{n}(N_{n}(L))\right|^{2}\right)=\mathbb{E}_{n}\left(\left|N_{n}^{0}(L)-\mathbb{E}_{n}(N_{n}^{0}(L))\right|^{2}\right).$

for simplicity set

$G_{k}(\gamma):=\left(\chi(\gamma^{k})+\overline{\chi(\gamma^{k})}\right)\cos(\alpha
k\ell(\gamma))\frac{\ell(\gamma)\widehat{\psi}(k\ell(\gamma)/L)}{2\sinh(k\ell(\gamma)/2)},$

so that we have

$\mathbb{V}_{n}(L)=$
$\frac{4}{L^{2}}\sum_{\gamma_{1},k_{1}}\sum_{\gamma_{2},k_{2}}G_{k_{1}}(\gamma_{1})\overline{G_{k_{2}}(\gamma_{2})}\mathbb{E}_{n}\left[\left(F_{n}(\gamma_{1}^{k_{1}})-\mathbb{E}_{n}(F_{n}(\gamma_{1}^{k_{1}}))\right)\left(F_{n}(\gamma_{2}^{k_{2}})-\mathbb{E}_{n}(F_{n}(\gamma_{2}^{k_{2}}))\right)\right].$

The parameter $L$ being fixed, we observe by Proposition 3.1 $(1)+(2)$ that
whenever $\gamma_{1}\neq\gamma_{2}\in\mathcal{P}_{0}$ we have

$\lim_{n\rightarrow+\infty}\mathbb{E}_{n}\left[\left(F_{n}(\gamma_{1}^{k_{1}})-\mathbb{E}_{n}(F_{n}(\gamma_{1}^{k_{1}}))\right)\left(F_{n}(\gamma_{2}^{k_{2}})-\mathbb{E}_{n}(F_{n}(\gamma_{2}^{k_{2}}))\right)\right]=0,$

therefore by Proposition 3.1 $(3)$ we get

$\lim_{n\rightarrow+\infty}\mathbb{V}_{n}(L):=\mathcal{S}(L):=\frac{4}{L^{2}}\sum_{k_{1},k_{2}}\mathcal{V}(k_{1},k_{2})S_{k_{1},k_{2}}(L),$

where

$S_{k_{1},k_{2}}(L):=\sum_{\gamma\in\mathcal{P}_{0}}G_{k_{1}}(\gamma)\overline{G_{k_{2}}(\gamma)}.$

We have obtained a ”near diagonal” sum, and the goal of the remaining
subsections is to analyze the asymptotic behaviour as $L\rightarrow+\infty$ of
each term $S_{k_{1},k_{2}}(L)$.

###  5.3. The non-primitive contribution

The non-primitive contribution corresponds to the sum over integers
$k_{1},k_{2}$ such that $k_{1}+k_{3}\geq 3$. We warn the reader that in the
estimates below, the implied constants do not depend on $L,k_{1},k_{2}$ and
may change from line to line. We will therefore use for simplicity
Vinogradov’s notation $f(T)\ll g(T)$ which means that $f(T)=O(g(T))$ as
$T\rightarrow+\infty$. By a crude bound we have

$|S_{k_{1},k_{2}}(L)|\ll\sum_{\gamma\in\mathcal{P}_{0}}e^{-\frac{k_{1}+k_{2}}{2}\ell(\gamma)}\ell(\gamma)^{2}.$

so that it is actually uniformly bounded with respect to $L$. By a Sieltjes
integration by parts we have

$|S_{k_{1},k_{2}}(L)|\ll\int_{0}^{\infty}e^{-\frac{k_{1}+k_{2}}{2}u}\left(\frac{k_{1}+k_{2}}{2}u^{2}-2u\right)N^{0}(u)du$
$\ll(k_{1}+k_{2})\int_{0}^{\infty}e^{-(\frac{k_{1}+k_{2}}{2}-1)u}u^{2}du,$

by using a crude bound on $N^{0}(u)\leq Ce^{u}$. Using the fact that for all
$\alpha>0$ we have

$\int_{0}^{\infty}e^{-\alpha t}t^{2}dt=\frac{1}{\alpha^{3}},$

we get for all $k_{1}+k_{2}\geq 3$,

$|S_{k_{1},k_{2}}(L)|\ll\frac{1}{(k_{1}+k_{2})^{2}}.$

We are now ready to estimate

$S_{NP}(L):=\sum_{k_{1}+k_{2}\geq
3}\mathcal{V}(k_{1},k_{2})S_{k_{1},k_{2}}(L).$

We will show that $S_{NP}(L)\ll\log^{2}(L)$, following a computation that was
suggested to us by Zeev Rudnick. First observe that since $\widehat{\psi}$ has
compact support, the sum over $k_{1},k_{2}$ is limited to the range

$k_{1},k_{2}\ll L.$

Notice that we have also

$\mathcal{V}(k_{1},k_{2})\ll\sigma(\gcd(k_{1},k_{2})),$

where

$\sigma(n)=\sum_{d|n}d$

is the sum of divisors function. Hence we have

$S_{NP}(L)\ll\sum_{k_{1},k_{2}\ll
L}\frac{\sigma(\gcd(k_{1},k_{2}))}{(k_{1}+k_{2})^{2}}.$

By writing for $j=1,2$, $k_{j}=Dm_{j}$ with $\gcd(m_{1},m_{2})=1$, we get

$S_{NP}(L)\ll\sum_{D}\sum_{m_{1}D,m_{2}D\ll
L}\frac{\sigma(D)}{D^{2}(m_{1}+m_{2})^{2}}$ $\ll\left(\sum_{D\ll
L}\frac{\sigma(D)}{D^{2}}\right)\left(\sum_{m_{1},m_{2}\ll
L}\frac{1}{(m_{1}+m_{2})^{2}}\right).$

It is easy to check that

$\left(\sum_{m_{1},m_{2}\ll L}\frac{1}{(m_{1}+m_{2})^{2}}\right)\ll\log(L).$

On the other hand, we can write

$\sigma(D)=\sum_{ab=D}a$

so that

$\sum_{D\ll L}\frac{\sigma(D)}{D^{2}}\ll\sum_{a,b\ll
L}\frac{a}{(ab)^{2}}\ll\sum_{a\ll
L}\frac{1}{a}\sum_{b=1}^{\infty}\frac{1}{b^{2}}\ll\log(L).$

In a nutshell, we have reached

$\lim_{n\rightarrow+\infty}\mathbb{V}_{n}(L)=\mathcal{S}(L)=\frac{4}{L^{2}}\sum_{k_{1},k_{2}}\mathcal{V}(k_{1},k_{2})S_{k_{1},k_{2}}(L)$
$=\frac{4}{L^{2}}S_{1,1}(L)+O\left(\frac{(\log(L))^{2}}{L^{2}}\right),$

where

$S_{1,1}(L)=\sum_{\gamma\in\mathcal{P}_{0}}\left(\chi(\gamma)+\overline{\chi(\gamma)}\right)^{2}\frac{\ell(\gamma)^{2}\widehat{\psi}^{2}(\ell(\gamma)/L)}{4\sinh^{2}(\ell(\gamma)/2)}\cos^{2}(\alpha\ell(\gamma)).$

###  5.4. Applying the equidistribution Lemma

We now know that

$\lim_{L\rightarrow+\infty}\lim_{n\rightarrow+\infty}\mathbb{V}_{n}(L)=\lim_{L\rightarrow+\infty}\frac{4}{L^{2}}S_{1,1}(L),$

thus we need to determine the asymptotic behaviour when $L\rightarrow+\infty$
of $S_{1,1}(L)$. We assume that $\alpha\neq 0$. We will therefore apply Lemma
5.1 with

$\Pi(x)=\sum_{\gamma\in\mathcal{P}_{0}\atop\ell(\gamma)\leq
x}\left(\chi(\gamma)+\overline{\chi(\gamma)}\right)^{2},\
g(x)=\frac{x^{2}}{4\sinh^{2}(x/2)}.$

Writing

$\Pi(x)=\sum_{\gamma\in\mathcal{P}_{0}\atop\ell(\gamma)\leq
x}\chi^{2}(\gamma)+\sum_{\gamma\in\mathcal{P}_{0}\atop\ell(\gamma)\leq
x}\overline{\chi}^{2}(\gamma)+\sum_{\gamma\in\mathcal{P}_{0}\atop\ell(\gamma)\leq
x}2,$

we observe that two cases can occur. Either $\chi^{2}\equiv 1$ and we have by
the prime orbit theorem (recall that we are summing over $\mathcal{P}_{0}$)

$\Pi(x)\sim 2\frac{e^{x}}{x},$

which by applying Lemma 5.1 gives

$\lim_{L\rightarrow+\infty}\frac{4}{L^{2}}S_{1,1}(L)=4\int_{0}^{\infty}u\widehat{\psi}^{2}(u)du=2\int_{\mathbb{R}}|u|\left(\widehat{\psi}(u))\right)^{2}du=\Sigma^{2}_{\mathrm{GOE}}(\psi).$

Or we have $\chi^{2}\not\equiv 1$ and we can apply Proposition 4.2 to get as
$x\rightarrow+\infty$

$\Pi(x)=\frac{e^{x}}{x}+O\left(\frac{e^{x}}{x^{2}}\right)+O\left(e^{\beta_{0}x}\right),$

for some $\beta_{0}<1$. Therefore as $x\rightarrow+\infty$ we have

$\Pi(x)\sim\frac{e^{x}}{x},$

and applying Lemma 5.1 gives now

$\lim_{L\rightarrow+\infty}\frac{4}{L^{2}}S_{1,1}(L)=2\int_{0}^{\infty}u\widehat{\psi}^{2}(u)du=\int_{\mathbb{R}}|u|\left(\widehat{\psi}(u))\right)^{2}du=\Sigma^{2}_{\mathrm{GUE}}(\psi).$

Theorem 1.1 is proved. We now show how Theorem 1.2 follows. In view of the
above dicussion, all we have to do is to compute the asymptotics of

$\Pi(x)=\sum_{\gamma\in\mathcal{P}_{0}\atop\ell(\gamma)\leq
x}\left(\mathrm{Tr}(\rho(\gamma))+\overline{\mathrm{Tr}(\rho(\gamma))}\right)^{2}.$

We can apply Proposition 4.3 with $\mathbf{G}$ as in Theorem 1.2 and

$f(g)=\left(\mathrm{Tr}(g)+\overline{\mathrm{Tr}(g)}\right)^{2}$

which says that

$\Pi(x)\sim\frac{1}{2}\left(\int_{\mathbf{G}}f(g)dg\right)\frac{e^{x}}{x}.$

Assume that $\mathbf{G}\subset U(N)$. First we can observe that

$\int_{\mathbf{G}}f(g)dg=\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg+\overline{\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg}+2\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg.$

As done previously in $\S 4$, let $\mathcal{I}:\mathbf{G}\rightarrow
GL(\mathbb{C}^{N})$ denote the standard representation of $\mathbf{G}$ acting
on $\mathbb{C}^{N}$ by left matrix multiplication. Since $\mathbf{G}$ is not a
direct product, $\mathcal{I}$ is actually irreducible, hence Schur
orthogonality says that

$\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg=\int_{\mathbf{G}}|\mathrm{Tr}(\mathcal{I}(g))|^{2}dg=1.$

On the other hand,

$\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg=\int_{\mathbf{G}}\mathrm{Tr}(\mathcal{I}\otimes\mathcal{I}(g))dg=d_{\mathrm{trivial}},$

where $d_{\mathrm{trivial}}$ denotes the multiplicity of the trivial
representation in the irreducible decomposition of
$\mathcal{I}\otimes\mathcal{I}$, in particular it is a non-negative integer.
Since

$\left|\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg\right|\leq\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg=1,$

we must have $d_{\mathrm{trivial}}\in\\{0,1\\}$. To summarize,

$\int_{\mathbf{G}}f(g)dg=2\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg+2\in\\{2,4\\},$

which shows that the only possible limits of the spectral variance are
$\Sigma^{2}_{\mathrm{GOE}}$ and $\Sigma^{2}_{\mathrm{GUE}}$. Since

$\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg\in\mathbb{R},$

we have

$0\leq\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg=\int_{\mathbf{G}}({\rm
Re}(\mathrm{Tr}(g)))^{2}dg-\int_{\mathbf{G}}({\rm
Im}(\mathrm{Tr}(g)))^{2}dg\leq\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg=1.$

Clearly if $g\mapsto\mathrm{Tr}(g)\in\mathbb{R}$, then

$\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg=\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg=1,$

and we are in the GOE case. If there exists $g\in\mathbf{G}$ such that ${\rm
Im}(\mathrm{Tr}(g))\neq 0$, then a simple continuity argument shows that

$\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg<\int_{\mathbf{G}}|\mathrm{Tr}(g)|^{2}dg=1,$

and therefore

$\int_{\mathbf{G}}(\mathrm{Tr}(g))^{2}dg=0,$

we are in the GUE case. Applying this criterion to the classical groups shows
that $\mathbf{G}=SO(N),Sp(N)$ will fall in the GOE case since traces are
always real. On the other hand, for $\mathbf{G}=U(N)$ and $\mathbf{G}=SU(N)$
with $N\geq 3$, traces can take complex values and we get the GUE case. Notice
that $Sp(1)=SU(2)$.

Theorem 1.2 is now proved. $\square$.

The reader might wonder if representations $\rho:\Gamma\rightarrow\mathbf{G}$
with Zariski dense images actually exist. A paper of Kishore [11], using
results of Breuillard, Green, Gurnalik and Tao, shows that if $\mathbf{G}$ is
real algebraic and semi-simple, then (Theorem 2.7), such a homomorphism always
exists. This is enough to deal with $\mathbf{G}=SU(N),SO(N),SP(N)$. More
Generally, in [10], they show (Corollary 2.4) using a dimensional argument,
that for any real algebraic reductive Lie Group $\mathbf{G}$, the set of
representations with Zariski dense image is non-empty (and actually dense)
provided that

$g\geq\left(\mathrm{dim}_{\mathbb{R}}(\mathbf{G})\right)^{2},$

where $g$ is the genus of $\Gamma\backslash\mathbb{H}$ and
$\mathrm{dim}_{\mathbb{R}}$ is the real dimension of $\mathbf{G}$. This allows
to build representations with zariski dense images in $U(N)$ provided $g\geq
N^{4}$.

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∎

11institutetext: B. Milholland22institutetext: Department of Genetics
Albert Einstein College of Medicine
1300 Morris Park Avenue
Bronx, NY 10461
USA
Tel.: +1 (718)-430-2000
22email<EMAIL_ADDRESS>
# The shortness of human life constitutes its limit

Brandon Milholland  Xiao Dong  Jan Vijg

(Received: date / Accepted: date)

###### Abstract

In this paper, we affirm our earlier findings of evidence for a limit to human
lifespan. In particular, we assess the analyses in extreme value theory (EVT)
performed by Rootzén and Zholud. We find that their criticisms of our work are
unfounded and that their analyses are contradicted by several other papers
using EVT. Furthermore, we find that even if we completely accept the
conclusions about late-life human mortality reached by Rootzén and Zholud,
their results do not actually contradict the findings presented in our
original paper: whether unbounded or not, human lifespan is unlikely to
greatly exceed 120 years, and the improbability of longer survival—whether it
is exactly zero or merely astronomically small—acts as a de facto limit. In
order to eliminate the confusion surrounding the issue, we propose the
adoption of the term “limit” to denote the age at which the chance of survival
is exactly zero and the term “effective limit” to denote the age at which the
change of survival falls below a given threshold. Once this distinction is
made, it can be demonstrated that the final result of Rootzén and Zholud is
essentially a recapitulation of the main conclusion of our paper. Ultimately,
much of the controversy surrounding the issue of a limit to human lifespan can
be avoided by carefully reading the literature and applying statistics to
practical human scales.

###### Keywords:

Extreme human life lengths Future record ages Supercentenarians Jeanne Calment
Limit for human life span Force of mortality IDL GRG HMD MRAD Evidence for a
limit to human lifespan Maximum lifespan Effective maximum lifespan

##  1 Introduction

In their paper rootzen_human_2017 , Rootzén and Zholud assert that there is no
limit to human lifespan and also level several criticisms at our paper
dong_evidence_2016  in which we reported evidence for such a limit. Harsh
words, such as “unfounded”, “inappropriate”, “wrong” and “misleading” are
used, but the severity of their language belies the fact that their paper is
the one that is misleading by misrepresenting our findings and attacking us on
tenuous, incorrect and unfounded bases (Section 2). A close examination of the
work of Rootzén and Zholud will reveal that their analysis is inadequate to
conclusively demonstrate that human lifespan is unlimited (Subsection 3.1).
Furthermore, several other analyses have come to conclusions the opposite of
Rootzén and Zholud (Subsection 3.2). Finally, even Rootzén and Zholud’s own
work suggests that there is a de facto limit to human lifespan, one that is
not much higher than the one we found in our original paper (Section 4). In
one very narrow sense—one that we did not propose at all in our paper—human
lifespan may be unlimited. But, in a much more practical and reasonable sense,
human lifespan, given the current paradigm of medical research, is limited.

##  2 Refuting the criticisms of Rootzén and Zholud

In Section 4 of their paper, Rootzén and Zholud summarily dismiss the the
results presented in our paper poiting to a limit to human lifespan
dong_evidence_2016  in four brief paragraphs111Rootzén and Zholud also
cursorily mention our findings that improvements in mortality decrease with
age and argue that “ a slower rate of improvement is still an improvement,
and, if anything, it contradicts the existence of a limit”, but this assertion
is never backed up by any evidence or reasoning. A slower rate of increase
does not imply that the increase will continue forever, nor does it imply that
there is no limit. By Rootzén and Zholud’s logic, an examination of some
partial sums of the series $\sum_{i=1}^{n}2^{-i}$ would show a slowing rate of
increase as $n$ increases, which “if anything … contradicts the existence of a
limit”. However, it has been mathematically proven that geometric series in
this form converge (i.e., $\lim_{n\to\infty}\sum_{i=1}^{n}2^{-i}=1$), so there
is in fact a finite limit to the sum. Furthermore, Rootzén and Zholud do not
address the fact that that there has been not merely a deceleration of
improvement but an absence of any detectable improvement whatsoever in
supercentenarian mortality.. These paragraphs only address Figure 2 of our
paper and do not address the findings presented in Figure 1 and Extended Data
Figures 1, 2, 3, 4, 5 and 6 of our paper. In these four crucial paragraphs,
Rootzén and Zholud begin with a falsehood, move on to a potential self-
contradiction, make a nonsensical statement, and finally conclude by
fundamentally misunderstanding the thesis of our paper.

In the first paragraph, Rootzén and Zholud state that the number of
supercentenarian deaths in the International Database of Longevity (IDL)
varies from 0 to 42. However, this statement is false: it is plain to see from
Rootzén and Zholud’s Figure 6 and Figure 7 that the minimum number of deaths
in any year is 1.

Rootzén and Zholud also assert in the first paragraph of Section 4 that the
yearly maximum reported age at death (MRAD) “shows the same pattern” as the
number of deaths; unfortunately, they do not provide any correlation
statistics that could be used to evaluate that assertion. In the next
paragraph, they mention a different line (representing $110+$ mean of maximum
$n_{t}$ exponential variables) that they also assert, without statistical
substantiation, “agree[s] well” with the MRAD, but this line differs greatly
from the line representing the number of deaths (most visible in the interval
1990–2000). It would seem contradictory that the line representing number of
deaths, which fluctuates in magnitude several-fold over short periods of time
(for example, leaping from 5 to 35 in the early 1990s), and the line
representing $110+$ mean of maximum $n_{t}$ exponential variables, which is
devoid of such wild fluctuations (barely ambling from around 113 to around 115
over the same interval), could both strongly correlate with the MRAD. Perhaps
the graphs are misleading, and both variables, counterintuitively, manage to
correlate well with the MRAD, but Rootzén and Zholud provide no statistical
evidence that they do.

In the third paragraph, Rootzén and Zholud say that our findings result from
“inappropriate combination of data from different time periods”. This does not
make sense: we merely used all the data from the entire time interval of the
IDL (and, separately, the GRG). Presumably, they meant to say that there was
an inappropriate combination of data from different countries since that is
the issue they touch upon at the end of the previous paragraph. However,
considering each country separately still finds evidence for a trend break,
contrary to Rootzén and Zholud’s assertion. In Japan, all of the data is post-
breakpoint, so there is no evidence for or against, although the number of
supercentenarian deaths does increase more rapidly than the MRAD, suggesting
the two have become decoupled. In the USA, the increase of the MRAD continues
unabated, providing some evidence against a trend break; but note also the
sharp jump in the number of supercentenarian deaths in the early 1990s and the
comparatively shallower increase in MRAD, again undermining Rootzén and
Zholud’s thesis that the MRAD is driven by number of supercentenarian deaths.
In England and Wales, the slope of the trend decreases from 0.7 to 0.2.
Rootzén and Zholud do not define what evidence is sufficient to indicate a
trend break, prompting the question: if a more than 3-fold decrease in the
slope is not evidence for a trend break, then what is? Finally, in France the
slope goes from 0.2 to –0.3, indisputable evidence of a trend break. To
summarize, of the four countries considered, one of them is invalid, one
arguably does not have a trend break, and two almost certainly do.
Furthermore, all four contain fluctuations in the number of supercentenarian
deaths not reflected in the MRAD. Therefore, the balance of evidence favors
the conclusions presented in our original paper.

The final paragraph dismisses the evidence of a limit in the 2nd through 5th
highest reported ages at death with a repetition of the nonsensical accusation
of “inappropriate combination of data from different time periods” and no
substantiating evidence. Given the weakness of the reasoning employed by
Rootzén and Zholud regarding the MRAD, it is likely that this assertion
regarding the 2nd through 5th highest deaths is not supported by the evidence,
but their thesis is ultimately impossible to evaluate due to the absence of
information provided in the paper. Section 4 of Rootzén and Zholud’s paper
terminates with a fundamental misinterpretation of the thesis of our paper.
The main observation of our paper was that the MRAD had stagnated since the
mid-1990s and, despite advances in medicine, fluctuated around 115, without
further increase; this stagnation around the average of 115 constitutes the
“limit” referred to in our paper. As a corollary, if one desires to know the
absolute highest age to which any human could reasonably expect to live, then
we also calculated that an MRAD of more than 125 would be expected only once
every 10,000 years; this value may also be considered a “limit” to human
longevity. In short, the term “limit” is overloaded, and can refer to
distinct, yet related, concepts depending on context. In their paper, Rootzén
and Zholud introduce a third definition of “limit”: the point at which the
chance of survival is equal to zero. They contend that since the distribution
of MRAD ages is unbounded, there is no “limit” to human lifespan. On its own,
there is nothing fundamentally wrong with this contention, but the failure of
our paper to conform to the arbitrarily altered definition of “limit”
formulated by Rootzén and Zholud cannot be considered an error on our part.
Indeed, as we will see, the definition of “limit” used by Rootzén and Zholud
is not very useful, as it would apply even to situations in which the
probability of survival is so small as to be negligible. From a very
philosophical perspective, it is possible to argue that due to the inherently
stochastic nature of the universe implied by quantum physics, it is impossible
to ever assign a probability of zero to any event, so the conclusion reached
by Rootzén and Zholud is not novel. Rootzén and Zholud build their paper on a
more immediately applicable starting point—demographic data—but if their goal
was to derive the conclusion of no limit to lifespan based on that foundation
alone, they have made several omissions and questionable decisions that
undermine the strength of their analyses.

##  3 Problems with the analyses of Rootzén and Zholud and literature
contradicting their results

Having made our reply to the criticisms Rootzén and Zholud leveled against our
paper, we move on to several of our own criticisms of their paper. The
conclusion reached by Rootzén and Zholud—that human life is “unlimited”—is
extraordinary, even if qualified by “but short”. Such a strong conclusion
requires strong evidence, but that which is presented by Rootzén and Zholud is
not up to the task. Furthermore, the conclusions reached by Rootzén and Zholud
put them at odds with multiple other papers, not just our own. These
contradictions must be resolved before the contention of an unlimited human
lifespan can be accepted.

###  3.1 Problems with the analyses of Rootzén and Zholud

Much of the work in Rootzén and Zholud’s paper is of good quality, but there
are several areas where they use statistics inappropriately or come to
conclusions that do not necessarily follow from the premises given. These
deficiencies do not discredit the entire paper, but they significantly weaken
the strength of the findings presented therein. Indeed, it is possible that
each could be addressed and a confirmation of Rootzén and Zholud’s analysis
found. Therefore, we present these issues in the spirit of a scientific
dialogue, in the hopes that further investigation will advance our knowledge,
either for or against Rootzén and Zholud’s conclusions.

First, the data used by Rootzén and Zholud appear to support a stagnation of
supercentenarian mortality, with no improvement in supercentenarian survival
in recent decades, something that would be consistent with a limit to human
lifespan and not a continual improvement towards breaking that limit. In Table
3 of their paper, Rootzén and Zholud present the results of statistical tests
in several sets of data for differences in supercentenarian mortality between
the first half of each set (generally spanning from the 1960s until the 1990s)
and the second half (generally from the mid-to-late 1990s until the mid-to-
late 2000s). In every single case, they find no evidence of a change in
mortality, with the lowest $p$-value being 0.18, not even close to
significant. This finding essentially confirms our earlier
resultsdong_evidence_2016 , i.e. a lack of improvement in supercentenarian
mortality indicates that human lifespan has stopped increasing, and is
essentially a restatement of the finding presented in Figure 3c of our
original paper.

Second, Rootzén and Zholud have chosen an inappropriate null hypothesis and
incorrectly interpreted the results of their statistical tests. Rootzén and
Zholud begin with the assumption that the force of mortality after age 110 is
constant, and that human lifespan therefore follows an exponential
distribution. Then, they provide several $p$-values, in Table 5 of their
paper, which fail to reject this null hypothesis. However, it is basic
knowledge regarding $p$-values that a failure to reject the null hypothesis
does not mean that the null hypothesis is true. Furthermore, the assumption of
a constant force of mortality is not an appropriate null hypothesis. From
early adulthood onwards, mortality increases relentlessly; it is far more
parsimonious to assume that this increase will continue rather than that it
will stop in its tracks at old age. Indeed, by making a constant force of
mortality (which corresponds to an “unlimited but short” lifespan) their null
hypothesis, Rootzén and Zholud have committed the fallacy of assuming that
which was to be demonstrated. Finally, despite having tilted the scales in
favor of a constant force of mortality, Table 2 of Rootzén and Zholud’s paper
indicates that the GRG dataset, the data with the greatest temporal and
geographic reach, strongly rejects the hypothesis of an exponential
distribution. Thus, the main premise of Rootzén and Zholud’s paper, that the
force of mortality after age 110 is constant, is not supported by the data and
statistics presented.

###  3.2 Contradictions with the literature

Having established that Rootzén and Zholud failed to demonstrate a constant
force of mortality, we should not be too harsh. After all, the task they have
set out for themselves is impossible: showing that the force of mortality is
constant would require showing that there is no change in mortality, which
would require proving a negative. A more realistic goal would be to calculate
a confidence interval for the force of mortality after age 110 in order to
establish an upper bound for mortality. That has been done by
othersmodig_how_2017 , and the upper bound of the resulting interval rapidly
approached 1. Although the confidence intervals were wide, leading the authors
of that paper to conclude that mortality after age 100 plateaued, the fact
that the estimate of mortality also rapidly increased past 0.5 after age 110
and rapidly approached 1 indicates that the balance of evidence is actually in
favor of a limited lifespan as defined by Rootzén and Zholud. Furthermore,
they found that centenarian mortality has not improved, a result that strongly
supports our findings of a limited human lifespan based on MRAD. As the
authors of that paper concluded: “the maximum lifespan, measured as the age of
the oldest person to die, is currently not increasing”.

Rootzén and Zholud have used EVT in their attempt to resolve the question of
whether human lifespan is limited, reaching the conclusion that it is
unlimited. Three other recent papers have examined the same question using EVT
and come to the opposite conclusion. The firstgbari_extreme_2017  examined
data from 46,666 Belgians who died at 95 or older and found an “ultimate age”
of 115 for men and 123 for women, a result very similar to our own. The second
paperfeifel_who_2017  analyzed data from the International Database on
Longevity (IDL) and the Human Mortality Database (HMD). The authors of that
paper write that “due to its small sample size, the hypothesis of an infinite
lifespan could not be rejected for the IDL,” the dataset used by Rootzén and
Zholud, but when they supplemented it with data from the HMD, “we found
significant evidence for a finite lifespan in the combined data set and
obtained reliable point estimates for the maximum attainable age” of 125–128
years in females. Finally, a papereinmahl_limits_2017  based on precise
measurements of the ages at death of 285,000 residents of the Netherlands
found “compelling statistical evidence” for a finite human lifespan in both
men and women. They found an average annual endpoint to the lifespan
distribution of 114 and 116 years and a maximum estimated upper endpoint of
125 and 124 years in men and women respectively222The authors of that paper
also found a lack of a trend in with time in these values; although they imply
that the lack of a trend would contradict our paper, this implication is based
on a misinterpretation of our results, which found an initial upward trend,
followed by a lack of a trend (and not necessarily a decrease, as indicated by
the insignificant $p$-value) during recent decades. Instead, the lack of a
trend is a confirmation of our findings that there has essentially been a
limit in place the whole time, which was finally reached in the mid-1990s. .
These estimates are very close to the estimates of average MRAD, 115 years,
and maximum MRAD, 125 years, which we arrived at in our original paper. To
summarize, all three of these other EVT papers confirm our results and
contradict those of Rootzén and Zholud.

To conclude this section, we would like to use Rootzén and Zholud’s concluding
sentences to tie together the two previous strands: the gaps in their paper
and its variance with the literature. In the penultimate and final sentences
of their paper, Rootzén and Zholud concede that “the IDL data set only
includes 9 persons who lived longer than 115 years, so data is sparse above
this age” but assert that “these 9 data points agree well with our
conclusion”. No substantiation is given for this assertion, and the very
sparsity of such data is itself suggestive of a limit to human lifespan, but
even if it were true that those 9 data points did support Rootzén and Zholud’s
conclusion, they must surely be outweighed by the contradictory studies based
on over 300,000 data points.

##  4 Shortness of life as a de facto limit: theoretical considerations and
empirical results

As we have shown, there is considerable evidence against Rootzén and Zholud’s
conclusion that the force of mortality remains constant after age 110.
However, there still remains a slim possibility that they are correct. It is
also possible that the force of mortality could increase with age and
asymptotically approach 1 without ever reaching it, such as in a sigmoid
function (Figure 1), a scenario not considered by Rootzén and Zholud. In any
case, let us assume that Rootzén and Zholud are correct there is no limit,
$L$, beyond which the probability of survival is 0. Would this contradict our
results? And would it be a finding of importance? The answer to both questions
is: no.

Figure 1: The three trajectories of late-life mortality envisioned by Rootzén
and Zholud, and one they do not consider, placed in context of the earlier
exponential increase in mortality. In the top left corner, the “limited”
scenario, mortality increases exponentially throughout life until it hits 1
(certain death). In the top right corner, the “unlimited but short” scenario,
mortality increases exponentially until it reaches 0.53 and then remains
stationary. In the bottom left corner, the “unlimited” scenario, mortality
increases exponentially and then decreases late in life. Finally, in the
bottom right corner, the sigmoid scenario not considered by Rootzén and
Zholud, mortality follows an approximately exponential increase early in life
until it begins to level off and asymptotically approach 1; mortality never
reaches 1, but it does not decline or stagnate and continues to increase at an
ever-slower rate. Note the lack of parsimony in both of the “unlimited”
scenarios proposed by Rootzén and Zholud.

First, a hard limit, with a chance of mortality equal to 1, was never part of
our original paper. As a commentary on our paper
observedolshansky_ageing:_2016 , our paper’s thesis is that “there is no fixed
limit beyond which humans cannot live, but that there are, nevertheless,
limits on the duration of life”; the “limit” referred not to an age of certain
death but the stagnation of advances in human maximum lifespan and the
corollary emergence of an age beyond which survival, although not impossible,
is prohibitively unlikely. To refer directly to the text of our paper: “we
found that the probability of an MRAD exceeding 125 in any given year is less
than 1 in 10,000”—not zero. In light of this, Rootzén and Zholud’s statement
that “it is likely that the record age …will be shorter than 128 years” is
neither contradictory to our conclusions nor particularly novel, but rather a
slightly different estimate of the number: stated informally (we shall
formalize it below), the age which MRAD is unlikely to exceed in the
foreseeable future.

The main thesis of Rootzén and Zholud’s paper, that there is no “limit” by
their definition to human lifespan is not particularly meaningful or novel.
Indeed, it is not even necessary to consult any demographic data in order to
arrive at the conclusion that there is no age with a zero chance of survival.
According to quantum physics, anything is possible, even if extremely
unlikely. So, the odds of a brain spontaneously appearing out of thin air due
to quantum fluctuations have been calculatedlinde_sinks_2007  to be $1$ in
$10^{10^{50}}$. Since aging is due to physical changes, i.e. the accumulation,
depletion or modification of macromoleculeslopez-otin_hallmarks_2013 , it is
possible that quantum fluctuations could spontaneously transform a
supercentenarian’s body to that of a 20 year old, making them a shoo-in for a
new longevity record. Calculating the exact probability of this event
occurring, and the question of whether the number of zeros in its decimal
representation would exceed the number of atoms in the observable universe, we
leave as an exercise for future investigators.

In case the scenario described above strikes readers as too outlandish, we
propose a more realistic occurrence based on Rootzén and Zholud’s model of
“unlimited” lifespan: assuming that Rootzén and Zholud are correct and
mortality after age 110 follows the “each year a coin is tossed” model, and
rounding333Observant readers will note that the abstract of Rootzén and
Zholud’s paper is contradicted by its body: in the former, the probability of
dying is given as 47%, while in the latter that value is given as the
probability of survival. While mixing up life and death is a major error, the
proximity of both probabilities to 50% limits its effects.  up the probability
of survival to 50%, the odds of a supercentenarian living to age 150 are $1$
in $2^{40}$ or just under 1 in a trillion. So, on the one hand we are told
that “human life is unlimited” and on the other hand the same model gives
probabilities of survival that are so low that many would consider them next
to impossible. At a certain point, discussing the probability of a single
individual living to a given age under Rootzén and Zholud’s “unlimited”
lifespan paradigm becomes akin to discussing the probability of receiving a
single molecule of the active ingredient from a homeopathic remedy.

Our point is that the finding that human lifespan is “unlimited” on the basis
that there is no age with a zero chance of survival may be mathematically true
and it may be of theoretical or even recreational interest, but it is of
little practical significance or use. Insisting that a “limit” to lifespan
consist of an age with 100% mortality gives the term “limit” too narrow and
restrictive a definition, one we did not use in our original paper and one
that makes any statements regarding its existence or nonexistence almost
completely inconsequential. Instead of discussing the limit, $L$, of lifespan
as the age with a zero chance of survival, we propose that future discussions
of extreme longevity focus on the effective limit, $L_{e}$ of lifespan, the
age beyond which it is extremely unlikely any individual of the species will
survive. The value of $L_{e}$ will depend on many factors, the most pressing
of which is the definition of “extremely unlikely”. Therefore, $L_{e}$ can be
stated formally by qualifying it with $\epsilon$, the desired definition of
“extremely unlikely”. For example, our original paper calculated
$L_{e}^{\epsilon=0.0001}=125$, i.e. the probability of any human surviving
past 125 is 0.0001 or 1 in 10,000. In their paper, Rootzén and Zholud
essentially calculate $L_{e}$ to be 128, but do not provide the value of
$\epsilon$ they used. The values of $L_{e}$ and $\epsilon$ can be inferred
from other publications (Table 1). Generally, $L_{e}^{\epsilon=0.5}$ is around
115, while values of $L_{e}$ for lower values of $\epsilon$ are in the
mid-120s.

Table 1: Values of $L_{e}$ and $\epsilon$ inferred from other publications.  $L_{e}$ | $\epsilon$ | Data source | Citation
---|---|---|---
114.9 | 0.5 | IDL/GRG | dong_evidence_2016
125 | 0.0001 | IDL/GRG | dong_evidence_2016
131.21 | $\ll{}0.025$ | Belgian National Population Register | gbari_extreme_2017
128.73 | $\ll{}0.025$ | IDL/HMD | feifel_who_2017
115.7 | 0.5 | Statistics Netherlands | einmahl_limits_2017
123.7 | 0.025 | Statistics Netherlands | einmahl_limits_2017
115 | 0.5 | “Best-guess” MRAD | milholland_best-guess_2017
128 | ? | IDL/GRG | rootzen_human_2017

The value of $L_{e}$, for a given value of $\epsilon$, is not constant. It is
determined by two factors: the population of the species (more individuals
means more chances for one of them to reach an extreme age) and the chance of
survival at each age leading up to $L_{e}$. In our original paper, we came to
the conclusion that $L_{e}$ had basically reached its limit—here, meaning that
$L_{e}$ had remained for years and would remain in the absence of
unprecedented technological breakthroughs at its current level—based on the
tacit assumption that the human population would not greatly increase and our
explicit demonstration that late-life survival had stagnated. Rootzén and
Zholud speculate that future research may further extend human lifespan, and
we do the same in the conclusion of our original paper, stating that “there is
no scientific reason why such efforts could not be successful”. However, the
gains from such efforts have not been realized. There is currently no cure for
aging, nor is there even a product approved to be marketed as a treatment for
aging. Perhaps one day human lifespan will become unlimited, but right now the
chances of mortality at old age act as an effective limit to lifespan.

##  5 Conclusion

The premise and even the title of Rootzén and Zholud’s paper are self-
contradictory: “human life is unlimited — but short”. We resolve this
contradiction by positing that the shortness of human life constitutes its
limit. For a particular, narrow definition of “limit” it can indeed be said
that there is no limit to human lifespan, but for a more useful definition
there is one. If one finds it justified to say that human life is “unlimited”
because there is a constant 47% chance of survival each year, one must also
find it justified to say that roulette winnings are “unlimited” because there
is a constant 48.7% chance of doubling one’s money by betting on black every
time. It is easy to see the problem with relying on the latter reasoning for a
moneymaking strategy, so why would anyone accept the former reasoning for a
longevity strategy?

The use of an overly restrictive definition of “limit” (as detailed in Section
4) is not the only issue with Rootzén and Zholud’s paper. They also make
unfounded criticisms of our findings of evidence for a limit to human lifespan
(refuted in Section 2) and the evidence even for their modest conclusion is
weak, with both statistical issues undermining their findings (enumerated in
Subsection 3.1) and larger studies having contradictory results (reviewed in
Subsection 3.2). Thus, the contention that “human life is unlimited” is most
likely to be false.

Much of the controversy surrounding this issue could have been avoided by
carefully reading our paper and using the context of the word “limit” as we
used it to correctly infer its meaning in that paper. Imposing an
inappropriate definition of the word does not contribute to the literature, as
it seeks to rebut our paper for a thesis it does not actually advance.

However, there is some value to Rootzén and Zholud’s work, as it has drawn
attention to the necessity of explicitly formulating a definition of “limited”
that is useful for discussions of human lifespan. Although the evidence
available suggests that, contrary to Rootzén and Zholud’s conclusion, the risk
of mortality does not remain constant after age 110, there also is no age at
which mortality is certain. In that sense, then, human life is unlimited: for
an infinitely large cohort or for an infinitely many cohorts, there is no
maximum age beyond which no individual will live. However, for finitely-sized
cohorts, observed over a sensible interval of time (10,000 years, being a few
times longer than the duration of recorded civilization, is probably the
absolute maximum that can be seriously considered) the age-related elevation
of mortality acts to constrain the highest age that is likely to be observed,
with estimates of that constraint clustering around 125. In order to
distinguish between these two concepts, we proposed the term “effective limit”
to indicate the age at which the probability of a single individual surviving
is negligible (falling below a given threshold). We hope that this new term
will be useful in facilitating clear and meaningful discussion of longevity in
the future.

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