# Weyl’s Law for compact rank one symmetric spaces

Anupam Pal Choudhury  School of Mathematical Sciences, National Institute of
Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National
Institute, Khurda 752050, Odisha, India<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Sai Sriharsha Indukuri  Department of
Mathematics, Indian Institute of Technology, Bombay, Maharashtra- 400076,
India<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Ritwik Mukherjee
School of Mathematical Sciences, National Institute of Science Education and
Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Khurda
752050, Odisha, India<EMAIL_ADDRESS>
###### Abstract.

Weyl’s law is a fundamental result governing the asymptotic behaviour of the
eigenvalues of the Laplacian. It states that for a compact $d$ dimensional
manifold $M$ (without boundary), the eigenvalue counting function
$N_{M}(\lambda)$ has the following asymptotic growth

$N_{M}(\lambda)=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right)\lambda^{d}+O(\lambda^{d-1}).$

A natural question is: when is the error term sharp and when can it be
improved? It has been known for a long time that the error term is sharp for
the round sphere (since $1968$). In contrast, it has only recently been shown
(in $2019$) by Iosevich and Wyman that for the product of spheres, the error
term can be polynomially improved. They conjecture that a polynomial
improvement should be true for products in general.
In this paper we extend both these results to Compact Rank One Symmetric
Spaces (CROSSes). We show that for CROSSes, the error term is sharp.
Furthermore, we show that for a product of CROSSes, the error term can be
polynomially improved. This gives further evidence to the conjecture made by
Iosevich and Wyman.

###### 2020 Mathematics Subject Classification:

35B40

##  1\. Introduction

Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary
and let $\Delta_{M}$ be the Laplace-Beltrami operator (acting on functions).
Associated to this operator, we define the eigenvalue counting function

$\displaystyle N_{M}(\lambda)$ $\displaystyle:=\\#\\{\textnormal{eigenvalues
of $\Delta_{M}$}\leq\lambda^{2}\\}.$

In other words, $N_{M}(\lambda)$ is the number of eigenvalues of $\Delta_{M}$
less than or equal to $\lambda^{2}$ (counted with multiplicities). It is a
fundamental result that the asymptotic growth of $N_{M}(\lambda)$ is governed
by Weyl’s law:

$\displaystyle\lim_{\lambda\longrightarrow\infty}\frac{N_{M}(\lambda)}{\lambda^{d}}$
$\displaystyle=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right),$
(1)

where $\omega_{d}$ is the volume of the unit ball in $\mathbb{R}^{d}$. A proof
of (1) can be found in [2, page 95, corollary 2.43]; it uses the the
asymptotic expansion of the heat kernel.
As with any question on asymptotics, it is natural to ask what can be said
about the error term. In $1920$, Richard Courant showed that the error term is
of the order of $O(\lambda^{d-1}\log(\lambda))$ ([5]). Since then, an
extensive amount of work has been done by mathematicians to study the error
term.
It was shown by Lars Hörmander in $1968$ ([7]) that the error term is
$O(\lambda^{d-1})$. He further showed in that paper that the
$O(\lambda^{d-1})$ error term is sharp for the round sphere $S^{d}$ (the
special case of the three dimensional sphere $S^{3}$ was worked out earlier by
Avacumovič in [1]).
Although the example of the sphere shows that the error, in general, can not
be made better than $O(\lambda^{d-1})$, it is still natural to ask under what
special cases the error can be improved. In [6] and [9], Duistermaat and
Guillemin and Ivrii show that if the measure of all periodic geodesic
billiards is zero, then the error term can be improved (see [8, page 595,
theorem 1.3] for the exact statement).
In [8], Iosevich and Wyman show [8, page 596, theorem 1.4] that for a product
of at least two compact Riemannian manifolds without boundary, the hypothesis
of Duistermaat and Guillemin and Ivrii’s Theorem holds; hence the error term
can be improved. But that is not all, they show that for the product of
spheres, the error term can be polynomially improved. More precisely, they
show [8, page 596, theorem 1.4] that if $M$ is the product of $n$ spheres,
then

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right)\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}).$
(2)

Based on this theorem, Iosevich and Wyman make the following conjecture.

######  Conjecture 1.1.

[8, page 597] Let $M:=M_{1}\times\ldots\times M_{n}$ be the product of at
least two compact Riemannian manifolds without boundary. Then there exists a
$\delta>0$ such that

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right)\lambda^{d}+O(\lambda^{d-1-\delta}).$
(3)

In this paper, we give further evidence to support the above conjecture.
Before stating the main result of this paper, let us recall from [3, page 76]
that a Compact Rank One Symmetric Spaces (CROSS) is either the sphere $S^{d}$,
the real projective space $\mathbb{RP}^{d}$, the complex projective space
$\mathbb{CP}^{d}$, the quaternionic projective space $\mathbb{HP}^{d}$ or the
two dimensional octonionic projective space $\mathbb{OP}^{2}$. Note that aside
from $S^{1}$ and $\mathbb{RP}^{1}$ (which is again $S^{1}$ with a different
radius), all the other CROSSes have real dimension at least two. The main
result of this paper is as follows.

######  Theorem 1.2.

Let $M:=M_{1}\times\ldots\times M_{n}$ be the product of at least two compact
Riemannian manifolds without boundary. Suppose each $M_{i}$ is one of the
CROSSes of real dimension at least two. Then

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right)\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}).$
(4)

Note that theorem 1.2 supports Conjecture 1.1, with $\delta:=\frac{n-1}{n+1}$.
The second result of our paper is as follows.

######  Theorem 1.3.

Let $M$ be one the $CROSSes$. Then the error term in Weyl’s law can not be
improved.

##  2\. Structure of the paper

In order to prove theorem 1.2, we use a fundamental lattice counting result
proved by Iosevich and Wyman [8, page 597, theorem 1.8]. The main observation
of this paper is that their lattice counting result applies to a much more
general setting than spheres; in particular it applies to all CROSSes.
In section 3, we describe the general setup in which we can prove theorem 1.2.
The approach will work if the eigenvalues and multiplicities are explicitly
given and a certain condition on the multiplicity function can be verified.
In section 5 we show that theorem 1.2 is true for product of all manifolds
whose multiplicity function satisfies the symmetry condition. In the proof, we
make extensive use of Iosevich and Wyman’s lattice counting result. In section
6, we verify that the symmetry condition is indeed satisfied for all the
CROSSes.
Finally in section 7, we show that the error is sharp for a single CROSS.

##  3\. Setup and notation

Our set up is as follows. Let $(M,g)$ be a compact Riemannian manifold without
boundary, with dimension $d\geq 2$. We consider the eigenvalues of the
Laplacian acting on functions. Assume that the (square root of) the
eigenvalues are given by

$\displaystyle\Lambda_{M}$
$\displaystyle:=\\{\sqrt{\alpha_{M}(k)}:k=0,1,2,\ldots\\},\qquad\textnormal{where}\qquad\alpha_{M}(k):=A_{M}k^{2}+B_{M}k+C_{M}.$
(5)

Here $A_{M},B_{M}$ and $C_{M}$ are constants depending on $M$. We will assume
that $A_{M}$ is not zero, i.e.

$\displaystyle A_{M}$ $\displaystyle>0.$ (6)

Assume that the multiplicity of these eigenvalues are given by a degree $d-1$
polynomial $\mathsf{R}_{M}(k)$. Define

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle:=\mathsf{R}_{M}\Big{(}t-\frac{B_{M}}{2A_{M}}\Big{)}.$ (7)

We now state the crucial assumption that the multiplicity function
$\mathsf{Q}_{M}$ should satisfy: we assume that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=Ct^{d-1}+\varphi(t),\qquad\textnormal{if}\qquad
d>2\qquad\textnormal{and}$ $\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=Ct,\qquad\textnormal{if}\qquad d=2,$ (8)

where $\varphi(t)$ is a polynomial of degree at most $d-3$ and $C$ is some
constant.

######  Definition 3.1.

Let $(M,g)$ be a compact Riemannian manifold without boundary, with dimension
$d\geq 2$. If conditions (5), (6) and (8) are satisfied, then we will say that
$(M,g)$ is a $\mathsf{W}$-manifold.

We have used the letter $\mathsf{W}$ in honour of Weyl.
The condition of being a $\mathsf{W}$-manifold will play a crucial role in the
proof of theorem 1.2. It will also play a role in the proof of theorem 1.3.
The main results of this paper can be rephrased by the following three
theorems.

######  Theorem 3.2.

Let $(M_{1},g_{1}),(M_{2},g_{2}),\ldots,(M_{n},g_{n})$ be
$\mathsf{W}$-manifolds of dimensions $d_{1},d_{2},\ldots,d_{n}$. Let

$\displaystyle M$ $\displaystyle:=M_{1}\times M_{2}\times\ldots\times
M_{n},\qquad\textnormal{and}\qquad d:=d_{1}+d_{2}\ldots+d_{n}.$

If $n>1$, then

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\textnormal{Vol}(M)\left(\frac{\omega_{d}}{(2\pi)^{d}}\right)\lambda^{d}+O\left(\lambda^{d-1-\frac{n-1}{n+1}}\right).$

######  Theorem 3.3.

Let $(M,g)$ be a $\mathsf{W}$-manifold. Then the error term in Weyl’s law can
not be improved.

######  Theorem 3.4.

Compact Rank One Symmetric Spaces of real dimension $d\geq 2$ are
$\mathsf{W}$-manifolds.

We note that theorems 3.2, 3.3 and 3.4 immediately imply theorems 1.2 and 1.3.

##  4\. A lattice counting result by Iosevich and Wyman

In this section, we recall the lattice counting theorem proved by Iosevich and
Wyman in [8].

######  Theorem 4.1.

[8, page 597, theorem 1.8] Fix $y\in\mathbb{R}^{n}$ and let
$(d_{1},d_{2},\dots,d_{k},1,\dots,1)\in\mathbb{N}^{n}$ be a multiindex with
$d_{i}\geq 2$ for $i=1,\dots,k$. Let $p$ be a norm on $\mathbb{R}^{n}$ such
that $p^{-1}(1)$ is a smooth hypersurface in $\mathbb{R}^{n}$ with
nonvanishing Gaussian curvature. Define $d:=d_{1}+\ldots+d_{k}+n-k$. For a
weight function

$F(x)=\prod_{i=1}^{k}\chi_{[0,\infty)}(x_{i})\ x_{i}^{d_{i}-1},$

we have

$\sum_{{m\in(\mathbb{Z}^{n}+y)}\atop{p(m)\leq\lambda}}F(m)=\lambda^{d}\int_{p^{-1}([0,1])}F(x)\
dx+E(\lambda),$

where $E(\lambda)$ satisfies the bounds

$E(\lambda)=O\left(\lambda^{d-1-\frac{n-1}{n+1}}\right)$

uniformly in $y$.

The following result is an immediate consequence of the previous theorem.

######  Theorem 4.2.

Let $d_{1},d_{2},\ldots,d_{n}\in\mathbb{Z}^{\geq 2}$ and $n\geq 2$. Let
$y_{1},\ldots,y_{n}$ be fixed $n$ real numbers and $P_{1},P_{2},\ldots P_{n}$
be fixed $n$ positive real numbers. Then

$\displaystyle\sum_{\begin{subarray}{c}m_{1}-y_{1},\ldots,m_{n}-y_{n}\in\mathbb{Z}^{\geq
0}\\\
P_{1}m_{1}^{2}+\ldots+P_{n}m_{n}^{2}\leq\lambda^{2}\end{subarray}}m_{1}^{d_{1}-1}\ldots
m_{n}^{d_{n}-1}$
$\displaystyle=C\lambda^{d}+O\left(\lambda^{d-1-\frac{n-1}{n+1}}\right),$ (9)

where $d:=d_{1}+d_{2}\ldots+d_{n}$ and $C$ is a constant.

Here, we use the norm $p$ defined on $\mathbb{R}^{n}$ by

$p(m):=\left(P_{1}m_{1}^{2}+\dots+P_{n}m_{n}^{2}\right)^{\frac{1}{2}},\
m=(m_{1},\dots,m_{n})\in\mathbb{R}^{n}.$

The crucial fact is that the ellipsoid $p^{-1}(1)$ has non vanishing Gaussian
curvature. Let us now use the above result to prove theorem 1.2 of this paper.

##  5\. Error can be polynomially improved for products of
$\mathsf{W}$-manifolds: proof

Let us first prove an auxiliary result. In the subsequent statements, $C$
always refers to a constant. We are going to make the standard abuse of
notation in where we refer to different constants by the same letter $C$.

######  Theorem 5.1.

Let $d_{1},d_{2},\ldots,d_{n}\in\mathbb{Z}^{\geq 2}$ and $n\geq 2$. let
$\mathsf{Q}_{d_{i}-1}(t)$ be a degree $d_{i}-1$ polynomial in $t$, that does
not have a $t^{d_{i}-2}$ term, i.e.

$\displaystyle\mathsf{Q}_{d_{i}-1}(t)$
$\displaystyle=C_{i}t^{d_{i}-1}+\varphi_{i}(t),\qquad\textnormal{if}\qquad
d_{i}>2\qquad\textnormal{and}$ $\displaystyle\mathsf{Q}_{d_{i}-1}(t)$
$\displaystyle=C_{i}t,\qquad\textnormal{if}\qquad d_{i}=2,$ (10)

where $C_{i}$ is a constant and $\varphi_{i}(t)$ is a polynomial of degree at
most $d_{i}-3$. Let $y_{1},\ldots,y_{n}$ be $n$ fixed real numbers and
$P_{1},P_{2},\ldots P_{n}$ be $n$ fixed positive real numbers. Given a real
number $\lambda$, let $N(\lambda)$ be defined as

$\displaystyle N(\lambda)$
$\displaystyle=\sum_{\begin{subarray}{c}m_{1}-y_{1},\ldots,m_{n}-y_{n}\in\mathbb{Z}^{\geq
0}\\\ P_{1}m_{1}^{2}+\ldots+P_{n}m_{n}^{2}\leq
a\lambda^{2}+b\end{subarray}}\mathsf{Q}_{d_{1}-1}(m_{1})\ldots\mathsf{Q}_{d_{n}-1}(m_{n}),$
(11)

where $a\in\mathbb{R}^{>0}$ and $b\in\mathbb{R}$. Then,

$\displaystyle N(\lambda)$
$\displaystyle=C\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}),$ (12)

where $d:=d_{1}+d_{2}\ldots+d_{n}$ and $C$ is a constant.

Proof: Let us unwind the summand of $N(\lambda)$. Let us denote the summation
of the leading power as $T_{1}$. The sum of the remaining powers will be
denoted by $T_{2}$. In other words,

$\displaystyle T_{1}$
$\displaystyle:=\sum_{\begin{subarray}{c}m_{1}-y_{1},\ldots,m_{n}-y_{n}\in\mathbb{Z}^{\geq
0}\\\ P_{1}m_{1}^{2}+\ldots+P_{n}m_{n}^{2}\leq
a\lambda^{2}+b\end{subarray}}Cm_{1}^{d_{1}-1}\ldots m_{n}^{d_{n}-1}.$ (13)

Note that the sum total of the powers of all the $m_{i}$ is $d-n$.
Next, we note that $T_{2}$ is a finite sum of the type

$\displaystyle\sum_{\begin{subarray}{c}m_{1}-y_{1},\ldots,m_{n}-y_{n}\in\mathbb{Z}^{\geq
0}\\\ P_{1}m_{1}^{2}+\ldots+P_{n}m_{n}^{2}\leq
a\lambda^{2}+b\end{subarray}}Cm_{1}^{e_{1}}\ldots m_{n}^{e_{n}},$ (14)

where the sum of the total power $e_{1}+\ldots e_{n}$ is at most $d-n-2$ (due
to the condition 10). In particular, the sum total of the powers of $m_{i}$ in
the summand of each of the terms of $T_{2}$ is at least $2$ less than the sum
total of the powers of the $m_{i}$ in the summand of $T_{1}$.
Using theorem 4.2, we conclude that

$\displaystyle T_{1}$
$\displaystyle=C\Big{(}a\lambda^{2}+b\Big{)}^{\frac{d}{2}}+O\left((a\lambda^{2}+b)^{\frac{1}{2}\left(d-1-\frac{n-1}{n+1}\right)}\right).$
(15)

Using binomial theorem, we conclude that

$\displaystyle\Big{(}a\lambda^{2}+b\Big{)}^{\frac{d}{2}}$
$\displaystyle=a^{\frac{d}{2}}\lambda^{d}+O(\lambda^{d-2}).$ (16)

Combining (15) and (16), we conclude that

$\displaystyle T_{1}$
$\displaystyle=C\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}).$ (17)

Similarly, we again apply theorem 4.2 and the binomial theorem to each of the
terms of $T_{2}$. Doing that, we conclude that $T_{2}$ will be given by the
right hand side of (17) with $d$ replaced by $d-2$, i.e.

$\displaystyle T_{2}$
$\displaystyle=C\lambda^{d-2}+O(\lambda^{d-3-\frac{n-1}{n+1}}).$ (18)

However, (18) implies that

$\displaystyle T_{2}$ $\displaystyle=O(\lambda^{d-2}).$ (19)

Since $N(\lambda)=T_{1}+T_{2}$, we conclude using (17) and (19), that

$\displaystyle N(\lambda)$
$\displaystyle=C\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}),$ (20)

thereby proving theorem 5.1. ∎
We are now ready to prove theorem 3.2.

Proof of theorem 3.2: Let $N_{M}(\lambda)$ denote the eigenvalue counting
function on $M$. Recall that $N_{M}(\lambda)$ can be expressed as

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\sum_{\begin{subarray}{c}k_{1},\ldots,k_{n}\in\mathbb{Z}^{\geq
0}\\\
\alpha_{M_{1}}(k_{1})+\ldots+\alpha_{M_{n}}(k_{n})\leq\lambda^{2}\end{subarray}}\mathsf{R}_{M_{1}}(k_{1})\ldots\mathsf{R}_{M_{n}}(k_{n}).$
(21)

Let us now make a substitution

$\displaystyle m_{1}$
$\displaystyle:=k_{1}+\frac{B_{M_{1}}}{2A_{M_{1}}},\ldots,m_{n}:=k_{n}+\frac{B_{M_{n}}}{2A_{M_{n}}}.$
(22)

Let $\mathsf{Q}_{M_{i}}(t)$ be as defined by (7), i.e.

$\displaystyle\mathsf{Q}_{M_{i}}(t)$
$\displaystyle:=\mathsf{R}_{M_{i}}\Big{(}t-\frac{B_{M_{i}}}{2A_{M_{i}}}\Big{)}.$
(23)

With this, the counting function becomes

$\displaystyle N_{M}(\lambda)$
$\displaystyle=\sum_{\begin{subarray}{c}m_{1}-\frac{B_{M_{1}}}{2A_{M_{1}}},\ldots,m_{n}-\frac{B_{M_{n}}}{2A_{M_{n}}}\in\mathbb{Z}^{\geq
0}\\\
A_{M_{1}}m_{1}^{2}+\ldots+A_{M_{n}}m_{n}^{2}\leq\lambda^{2}+\kappa\end{subarray}}\mathsf{Q}_{M_{1}}(m_{1})\ldots\mathsf{Q}_{M_{n}}(m_{n}),$
(24)

where $\kappa$ is given by

$\displaystyle\kappa$
$\displaystyle:=\sum_{i=1}^{n}\Big{(}\frac{B_{M_{i}}^{2}}{4A_{M_{i}}}-C_{M_{i}}\Big{)}.$

Since each of the $M_{i}$ is a $\mathsf{W}$-manifold, we can use theorem 5.1
and conclude that

$\displaystyle N_{M}(\lambda)$
$\displaystyle=C\lambda^{d}+O(\lambda^{d-1-\frac{n-1}{n+1}}).$ (25)

The constant $C$ can now be obtained from Weyl’s law for compact manifolds
without boundary.

##  6\. CROSSes are $\mathsf{W}$-manifolds: proof

We will now prove theorem 3.4, i.e. we will verify that all the CROSSes are
$\mathsf{W}$-manifolds.
First of all, we recall that the eigenvalues and multiplicities of the
Laplacian can be packaged together in the zeta function. If we are in the
setting of section 3, we get that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\mathsf{R}_{M}(k)\exp(-\alpha_{M}(k)s).$
(26)

We will now use the results of the paper [4], where the authors explicitly
compute $\zeta_{M}(s)$ for all CROSSes. Knowing $\zeta_{M}(s)$ immediately
tells us what is $\alpha_{M}(k)$ and $\mathsf{R}_{M}(k)$. Let us now
explicitly write down the zeta function for each of these spaces. Although the
case of the sphere has been worked out in [8], we will include it here for
completeness.

###  6.1. Odd dimensional spheres

Let us start with the odd dimensional sphere $M=S^{2d-1}$, with $d\geq 3$.
Following [4, page 7, theorem 2.4], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{k+d-1}{d-1}\prod_{j=1}^{2d-3}\frac{k+j}{j}\Big{)}\exp\Big{(}-\Big{(}\frac{k^{2}+2k(d-1)}{4(d-1)}\Big{)}s\Big{)}.$
(27)

Comparing (27) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\Big{(}\frac{k^{2}+2k(d-1)}{4(d-1)}\Big{)}\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=\Big{(}\frac{k+d-1}{d-1}\prod_{j=1}^{2d-3}\frac{k+j}{j}\Big{)}.$
(28)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2d-2$. Now,

$\alpha_{M}(k)=\frac{1}{4(d-1)}k^{2}+\frac{1}{2}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{4(d-1)},B_{M}=\frac{1}{2},C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=d-1.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}(t-(d-1))=\frac{t}{d-1}\prod_{j=1}^{2d-3}\frac{t-(d-1)+j}{j}=\frac{t}{d-1}\prod_{j=1}^{2d-3}\frac{t+a_{j}}{j},$
(29)

where $a_{j}:=j-(d-1),\ j=1,\dots,2d-3$.
We claim that the coefficient of $t^{2d-3}$ in $\mathsf{Q}_{M}(t)$ is zero. To
see this, we note that it is sufficient to prove that the numerator of the
coefficient of $t^{2d-4}$ in $\prod_{j=1}^{2d-3}\frac{t+a_{j}}{j}$, that is,
$\sum_{j=1}^{2d-3}a_{j}$ is zero. Now,

$\displaystyle\sum_{j=1}^{2d-3}a_{j}$
$\displaystyle=\sum_{j=1}^{2d-3}\left(j-d+1\right)=\frac{(2d-3)(2d-2)}{2}-(d-1)(2d-3)=0,$

and hence, we are done.
To finish the case of odd spheres, we also need to separately consider
$M=S^{3}$. Following [4, page 6, equation 2.3 (a)], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}(k+1)^{2}\exp\Big{(}-\Big{(}\frac{k^{2}+2k}{4}\Big{)}s\Big{)}.$
(30)

Comparing (30) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\Big{(}\frac{k^{2}+2k}{4}\Big{)}\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=(k+1)^{2}.$
(31)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2$. Now,

$\alpha_{M}(k)=\frac{1}{4}k^{2}+\frac{1}{2}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{4},B_{M}=\frac{1}{2},C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=1.$

Next, we apply definition (7) and conclude that the coefficient of $t$ in the
polynomial

$\displaystyle\mathsf{Q}_{M}(t)$ $\displaystyle=\mathsf{R}_{M}(t-1)$ (32)

is zero. This completes all the odd spheres.

###  6.2. Even dimensional spheres

Next, we consider even dimensional spheres $M=S^{2d}$, with $d\geq 1$.
Following [4, page 8, theorem 3.2], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{2k+2d-1}{2d-1}\prod_{j=1}^{2d-2}\frac{k+j}{j}\Big{)}\exp\Big{(}-\Big{(}\frac{k^{2}+k(2d-1)}{4d-2}\Big{)}s\Big{)}.$
(33)

Comparing (33) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\Big{(}\frac{k^{2}+k(2d-1)}{4d-2}\Big{)}\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=\Big{(}\frac{2k+2d-1}{2d-1}\prod_{j=1}^{2d-2}\frac{k+j}{j}\Big{)}.$
(34)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2d-1$. Now,

$\alpha_{M}(k)=\frac{1}{4d-2}k^{2}+\frac{2d-1}{4d-2}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{4d-2},B_{M}=\frac{1}{2},C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{2d-1}{2}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\Big{(}t-\frac{2d-1}{2}\Big{)}=\frac{2}{(2d-1)}t\cdot\prod_{j=1}^{2d-2}\frac{t+a_{j}}{j},$
(35)

where $a_{j}:=-\frac{2d-1}{2}+j,\ j=1,\dots,2d-2$.
We claim that the coefficient of $t^{2d-2}$ in $\mathsf{Q}_{M}(t)$ is zero. To
see this, we note that it is sufficient to prove that the numerator of the
coefficient of $t^{2d-3}$ in $\prod_{j=1}^{2d-2}\frac{t+a_{j}}{j}$, that is,
$\sum_{j=1}^{2d-2}a_{j}$ is zero. Now,

$\displaystyle\sum_{j=1}^{2d-2}a_{j}$
$\displaystyle=\sum_{j=1}^{2d-2}\left(j-\frac{2d-1}{2}\right)=\frac{(2d-2)(2d-1)}{2}-\frac{(2d-1)(2d-2)}{2}=0,$

and hence, we are done. This completes all the spheres.

###  6.3. Odd dimensional real projective spaces

Let us now move on to the real projective spaces. We will start with the odd
dimensional real projective spaces $M=\mathbb{RP}^{2d-1}$, with $d\geq 3$.
Following [4, page 8, corollary 2.6], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{2k+d-1}{d-1}\prod_{j=1}^{2d-3}\frac{2k+j}{j}\Big{)}\exp\Big{(}-\Big{(}\frac{k^{2}+k(d-1)}{d-1}\Big{)}s\Big{)}.$
(36)

Comparing (36) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\left(\frac{k^{2}+k(d-1)}{d-1}\right)\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=\left(\frac{2k+d-1}{d-1}\prod_{j=1}^{2d-3}\frac{2k+j}{j}\right).$
(37)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2d-2$. Now,

$\alpha_{M}(k)=\frac{1}{d-1}k^{2}+k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{d-1},B_{M}=1,C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{d-1}{2}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\Big{(}t-\frac{d-1}{2}\Big{)}=\frac{2t}{d-1}\prod_{j=1}^{2d-3}\frac{2t+a_{j}}{j},$
(38)

where $a_{j}:=j-(d-1),\ j=1,\dots,2d-3$.
We claim that the coefficient of $t^{2d-3}$ in $\mathsf{Q}_{M}(t)$ is zero. To
see this, we note that it is sufficient to prove that the numerator of the
coefficient of $t^{2d-4}$ in
$\prod_{j=1}^{2d-3}\frac{2t+a_{j}}{j}=\prod_{j=1}^{2d-3}\frac{t+\frac{a_{j}}{2}}{\frac{j}{2}}$,
that is, $\sum_{j=1}^{2d-3}\frac{a_{j}}{2}$ is zero. Now,

$\displaystyle\sum_{j=1}^{2d-3}\frac{a_{j}}{2}$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{2d-3}\left(j-d+1\right)=\frac{1}{2}\left(\frac{(2d-3)(2d-2)}{2}-(d-1)(2d-3)\right)=0,$

and hence, we are done.
To finish the case of the odd real projective spaces, we also need to
separately consider $M=\mathbb{RP}^{3}$. Following [4, page 6, equation 2.3
(b)], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}(2k+1)^{2}\exp\Big{(}-(k^{2}+k)s\Big{)}.$
(39)

Comparing (39) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=k^{2}+k\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=(2k+1)^{2}.$
(40)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2$. Now,

$\alpha_{M}(k)=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=1,B_{M}=1,C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{1}{2}.$

Next, we apply definition (7) and observe that the coefficient of $t$ in the
polynomial

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\Big{(}t-\frac{1}{2}\Big{)}=4t^{2}$ (41)

is zero. This completes all the odd real projective spaces.

###  6.4. Even dimensional real projective spaces

Next, we consider even dimensional real projective space $M=\mathbb{RP}^{2d}$,
with $d\geq 1$. Following [4, pages 8-9, corollary 3.4], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{4k+2d-1}{2d-1}\prod_{j=1}^{2d-2}\frac{2k+j}{j}\Big{)}\exp\Big{(}-\Big{(}\frac{2k^{2}+k(2d-1)}{2d-1}\Big{)}s\Big{)}.$
(42)

Comparing (42) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\Big{(}\frac{2k^{2}+k(2d-1)}{2d-1}\Big{)}\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=\Big{(}\frac{4k+2d-1}{2d-1}\prod_{j=1}^{2d-2}\frac{2k+j}{j}\Big{)}.$
(43)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2d-1$. Now,

$\alpha_{M}(k)=\frac{2}{2d-1}k^{2}+k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{2}{2d-1},B_{M}=1,C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{2d-1}{4}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\Big{(}t-\frac{2d-1}{4}\Big{)}=\frac{4t}{2d-1}\prod_{j=1}^{2d-2}\frac{2t+a_{j}}{j},$
(44)

where $a_{j}:=j-\frac{2d-1}{2},\ j=1,\dots,2d-2$. We claim that the
coefficient of $t^{2d-2}$ in $\mathsf{Q}_{M}(t)$ is zero. To see this, we note
that it is sufficient to prove that the numerator of the coefficient of
$t^{2d-3}$ in
$\prod_{j=1}^{2d-2}\frac{2t+a_{j}}{j}=\prod_{j=1}^{2d-2}\frac{t+\frac{a_{j}}{2}}{\frac{j}{2}}$,
that is, $\sum_{j=1}^{2d-2}\frac{a_{j}}{2}$ is zero. Now,

$\displaystyle\sum_{j=1}^{2d-2}\frac{a_{j}}{2}$
$\displaystyle=\frac{1}{2}\sum_{j=1}^{2d-2}\left(j-\frac{2d-1}{2}\right)=\frac{1}{2}\left(\frac{(2d-2)(2d-1)}{2}-\frac{(2d-1)(2d-2)}{2}\right)=0,$

and hence, we are done.

###  6.5. Complex projective spaces

Let us now move on to the complex projective spaces $M=\mathbb{CP}^{d}$, with
$d\geq 2$. Notice that the real dimension of this manifold is $2d$. Following
[4, page 9, theorem 4.2], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{2k+d}{d}\prod_{j=1}^{d-1}\Big{(}\frac{k+j}{j}\Big{)}^{2}\Big{)}\exp\Big{(}-\Big{(}\frac{k^{2}+kd}{d+1}\Big{)}s\Big{)}.$
(45)

Comparing (45) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\left(\frac{k^{2}+kd}{d+1}\right)\qquad\textnormal{and}\qquad\mathsf{R}_{M}(k)=\left(\frac{2k+d}{d}\prod_{j=1}^{d-1}\Big{(}\frac{k+j}{j}\Big{)}^{2}\right).$
(46)

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $2d-1$. Now,

$\alpha_{M}(k)=\frac{1}{d+1}k^{2}+\frac{d}{d+1}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{d+1},B_{M}=\frac{d}{d+1},C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{d}{2}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\left(t-\frac{d}{2}\right)=\frac{2t}{d}\prod_{j=1}^{d-1}\left(\frac{t+a_{j}}{j}\right)^{2},$
(47)

where $a_{j}:=j-\frac{d}{2},\ j=1,\dots,d-1$. We claim that the coefficient of
$t^{2d-2}$ in $\mathsf{Q}_{M}(t)$ is zero. To see this, we note that it is
sufficient to prove that the numerator of the coefficient of $t^{2d-3}$ in
$\prod_{j=1}^{d-1}\left(\frac{t+a_{j}}{j}\right)^{2}$, that is,
$\sum_{j=1}^{d-1}2a_{j}$ is zero. Now,

$\displaystyle\sum_{j=1}^{d-1}2a_{j}$
$\displaystyle=2\sum_{j=1}^{d-1}\left(j-\frac{d}{2}\right)=2\left(\frac{d(d-1)}{2}-\frac{d(d-1)}{2}\right)=0,$

and hence, we are done.

###  6.6. Quaternionic projective spaces

Let us now move on to the quaternionic projective spaces
$M=\mathbb{HP}^{d-1}$, with $d\geq 2$. We are referring to the (quaternionic)
dimension as $d-1$ as opposed to $d$ in order to make it easy to follow the
notation of [4]. Notice that the real dimension of $\mathbb{HP}^{d-1}$ is
$4d-4$. Following [4, page 10, theorem 5.2], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\Big{(}\frac{2k+2d-1}{2d-1}\prod_{j=2}^{2d-2}\frac{k+j}{j}\prod_{l=1}^{2d-3}\frac{k+l}{l}\Big{)}\exp\Big{(}-\Big{(}\frac{k^{2}+2kd-k}{2(d+1)}\Big{)}s\Big{)}.$
(48)

Comparing (48) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\Big{(}\frac{k^{2}+2kd-k}{2(d+1)}\Big{)}\qquad\textnormal{and}$
$\displaystyle\mathsf{R}_{M}(k)$
$\displaystyle=\Big{(}\frac{2k+2d-1}{2d-1}\prod_{j=2}^{2d-2}\frac{k+j}{j}\prod_{l=1}^{2d-3}\frac{k+l}{l}\Big{)}.$

Notice that $\mathsf{R}_{M}(k)$ is a polynomial of degree $4d-5$. Now,

$\alpha_{M}(k)=\frac{1}{2(d+1)}k^{2}+\frac{2d-1}{2(d+1)}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{2(d+1)},B_{M}=\frac{2d-1}{2(d+1)},C_{M}=0$ and
therefore,

$\frac{B_{M}}{2A_{M}}=\frac{2d-1}{2}=d-\frac{1}{2}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\left(t-\left(d-\frac{1}{2}\right)\right)=\frac{2t}{2d-1}\prod_{j=2}^{2d-2}\frac{t+a_{j}}{j}\prod_{l=1}^{2d-3}\frac{t+b_{l}}{l},$
(49)

where

$\displaystyle a_{j}$ $\displaystyle=j-\left(d-\frac{1}{2}\right),\
j=2,\dots,2d-2,$ $\displaystyle b_{l}$
$\displaystyle=l-\left(d-\frac{1}{2}\right),\ l=1,\dots,2d-3.$

We claim that the coefficient of $t^{4d-6}$ in $\mathsf{Q}_{M}(t)$ is zero. To
see this, we note that it is sufficient to prove that the numerator of the
coefficient of $t^{4d-7}$ in
$\prod_{j=2}^{2d-2}\frac{t+a_{j}}{j}\prod_{l=1}^{2d-3}\frac{t+b_{l}}{l}$, that
is, $\sum_{j=2}^{2d-2}a_{j}+\sum_{l=1}^{2d-3}b_{l}$ is zero. Now,

$\displaystyle\sum_{j=2}^{2d-2}a_{j}+\sum_{l=1}^{2d-3}b_{l}$
$\displaystyle=\sum_{j=2}^{2d-2}j+\sum_{l=1}^{2d-3}l-(4d-6)\left(d-\frac{1}{2}\right)$
$\displaystyle=\left(\frac{(2d-2)(2d-1)}{2}-1\right)+\frac{(2d-3)(2d-2)}{2}-\frac{(4d-6)(2d-1)}{2}=0,$

and hence, the claim is true.

###  6.7. Octonionic projective plane

Finally, we consider $M=\mathbb{OP}^{2}$, the octonionic projective plane.
Note that the real dimension of this space is $16$. Following [4, page 11,
theorem 6.2], we conclude that

$\displaystyle\zeta_{M}(s)$
$\displaystyle=\sum_{k=0}^{\infty}\frac{2k+11}{11}\left(\prod_{j=1}^{3}\frac{k+j}{j}\right)\left(\prod_{l=4}^{7}\left(\frac{k+l}{l}\right)^{2}\right)\left(\prod_{m=8}^{10}\frac{k+m}{m}\right)\exp\left(-\frac{(k^{2}+11k)}{18}s\right).$
(50)

Comparing (50) with (26), we conclude that

$\displaystyle\alpha_{M}(k)$
$\displaystyle=\frac{(k^{2}+11k)}{18}\qquad\textnormal{and}$
$\displaystyle\mathsf{R}_{M}(k)$
$\displaystyle=\frac{2k+11}{11}\left(\prod_{j=1}^{3}\frac{k+j}{j}\right)\left(\prod_{l=4}^{7}\left(\frac{k+l}{l}\right)^{2}\right)\left(\prod_{m=8}^{10}\frac{k+m}{m}\right).$

Note that $\mathsf{R}_{M}(k)$ is a polynomial of degree $15$. Now,

$\alpha_{M}(k)=\frac{1}{18}k^{2}+\frac{11}{18}k=A_{M}k^{2}+B_{M}k+C_{M},$

where $A_{M}=\frac{1}{18},B_{M}=\frac{11}{18},C_{M}=0$ and therefore,

$\frac{B_{M}}{2A_{M}}=\frac{11}{2}.$

Next, we apply definition (7) and conclude that

$\displaystyle\mathsf{Q}_{M}(t)$
$\displaystyle=\mathsf{R}_{M}\Big{(}t-\frac{11}{2}\Big{)}=\frac{2t}{11}\prod_{j=1}^{3}\frac{t+a_{j}}{j}\prod_{l=4}^{7}\left(\frac{t+a_{l}}{l}\right)^{2}\prod_{m=8}^{10}\frac{t+a_{m}}{m},$
(51)

where

$\displaystyle a_{j}$ $\displaystyle=j-\frac{11}{2},\ j=1,\dots,10.$

We claim that the coefficient of $t^{14}$ in $\mathsf{Q}_{M}(t)$ is zero. To
see this, we note that it is sufficient to prove that the numerator of the
coefficient of $t^{13}$ in
$\prod_{j=1}^{3}\frac{t+a_{j}}{j}\prod_{l=4}^{7}\left(\frac{t+a_{l}}{l}\right)^{2}\prod_{m=8}^{10}\frac{t+a_{m}}{m}$,
that is, $\sum_{j=1}^{3}a_{j}+\sum_{l=4}^{7}2a_{l}+\sum_{m=8}^{10}a_{m}$ is
zero. Now,

$\displaystyle\sum_{j=1}^{3}a_{j}+\sum_{l=4}^{7}2a_{l}+\sum_{m=8}^{10}a_{m}$
$\displaystyle=\sum_{j=1}^{3}j+\sum_{m=8}^{10}m+2\sum_{l=4}^{7}l+3\left(-\frac{11}{2}\right)+3\left(-\frac{11}{2}\right)+8\left(-\frac{11}{2}\right)=0,$

and hence, the claim is proved.
This completes the proof of theorem 3.4. ∎

##  7\. Error is sharp for a single $\mathsf{W}$-manifold: proof

In this section, we will prove theorem 3.3. It was shown by Hörmander in [7]
that the error term in Weyl’s law can not be improved for the round sphere. We
will generalize his proof to $\mathsf{W}$-manifolds. Let us first prove a few
auxiliary results.

######  Proposition 7.1.

Let us define the following function
$\psi_{k}:\mathbb{N}\longrightarrow\mathbb{R}$, given by

$\displaystyle\psi_{k}(n)$ $\displaystyle:=\sum_{i=1}^{n}i^{k-1}$
$\displaystyle=1^{k-1}+2^{k-1}+\ldots+n^{k-1},$ (52)

where $k\in\mathbb{Z}^{\geq 2}$. Then,

$\displaystyle\psi_{k}(n)$
$\displaystyle=\frac{n^{k}}{k}+\frac{1}{2}n^{k-1}+\mathcal{R}_{k-2}(n),$ (53)

where $\mathcal{R}_{k-2}(n)$ is a polynomial in $n$ of degree at most $k-2$.

###### Proof.

Let $n\in\mathbb{N}$ and $k\in\mathbb{Z}^{\geq 2}$. Note that

$\displaystyle(n+1)^{k}-n^{k}=kn^{k-1}+\frac{k(k-1)}{2}n^{k-2}+\cdots+1$ (54)
$\displaystyle
n^{k}-(n-1)^{k}=k(n-1)^{k-1}+\frac{k(k-1)}{2}(n-1)^{k-2}+\cdots+1$
$\displaystyle\cdots$ $\displaystyle\cdots$ $\displaystyle 2^{k}-1^{k}=k\cdot
1^{k-1}+\frac{k(k-1)}{2}\cdot 1^{k-2}+\cdots+1.$

Adding up all the terms on the left hand side of equation (54), we get a
telescopic sum which equals $(n+1)^{k}-1^{k}$. Equating this with the
summation of the right hand side of equation (54), we conclude that

$(n+1)^{k}-1^{k}=k\psi_{k}(n)+\frac{k(k-1)}{2}\psi_{k-1}(n)+\cdots+\psi_{1}(n).$
(55)

In equation (55), we note that $\psi_{1}(n)=n$. Using the fact that
$\psi_{2}(n)=\frac{n(n+1)}{2}$ and equation (55), equation (53) follows by
using induction on $k$. This proves the desired result. ∎

######  Corollary 7.2.

Let us define the following function $N:\mathbb{R}\longrightarrow\mathbb{R}$,
given by

$\displaystyle N(\lambda)$ $\displaystyle:=\sum_{i=1}^{[\lambda]}i^{k-1}$
$\displaystyle=1^{k-1}+2^{k-1}+\ldots+[\lambda]^{k-1},$ (56)

where $k\in\mathbb{Z}^{\geq 2}$. Then,

$\displaystyle N(\lambda)$
$\displaystyle=\frac{1}{k}[\lambda]^{k}+\frac{1}{2}[\lambda]^{k-1}+\mathcal{R}_{k-2}([\lambda]),$

where $\mathcal{R}_{k-2}([\lambda])$ is a polynomial in $[\lambda]$ of degree
at most $k-2$.

###### Proof.

Follows immediately from proposition 7.1. ∎

Let us now make a couple of definitions, which will make it easier to state
the remaining results.

######  Definition 7.3.

Let $N,M:\mathbb{R}\rightarrow\mathbb{R}$. We will say that $N$ is
asymptotically equivalent to $M$ to first order (which will be written as
$N\sim_{1}M$) if there exists $d\in\mathbb{Z}^{\geq 1}$ such that

$\lim_{\lambda\rightarrow\infty}\frac{N(\lambda)}{\lambda^{d}}=c_{1}=\lim_{\lambda\rightarrow\infty}\frac{M(\lambda)}{\lambda^{d}},$

where $c_{1}$ is a non-zero real number.

######  Definition 7.4.

Let $N,M:\mathbb{R}\rightarrow\mathbb{R}$. We will say that $N$ is
asymptotically equivalent to $M$ to second order (which will be written as
$N\sim_{2}M$) if there exists $d\in\mathbb{Z}^{\geq 1}$ such that

$\lim_{\lambda\rightarrow\infty}\frac{N(\lambda)}{\lambda^{d}}=c_{1}=\lim_{\lambda\rightarrow\infty}\frac{M(\lambda)}{\lambda^{d}},$

and

$\lim_{\lambda\rightarrow\infty}\frac{N(\lambda)-c_{1}\lambda^{d}}{\lambda^{d-1}}=c_{2}=\lim_{\lambda\rightarrow\infty}\frac{M(\lambda)-c_{1}\lambda^{d}}{\lambda^{d-1}},$

where $c_{1},c_{2}$ are non-zero real numbers.

######  Proposition 7.5.

Let $c_{1}$ and $c_{2}$ be non-zero real numbers and $k\in\mathbb{Z}^{\geq
1}$. Define $N,M:\mathbb{R}\longrightarrow\mathbb{R}$ as

$\displaystyle N(\lambda)$
$\displaystyle:=c_{1}[\lambda]^{k}+c_{2}[\lambda]^{k-1}+\alpha([\lambda])\qquad\textnormal{and}\qquad
M(\lambda):=c_{1}\lambda^{k}+c_{2}\lambda^{k-1}+\beta(\lambda),$

where $\alpha$ and $\beta$ are polynomials of degree at most $k-2$ if $k\geq
2$ and they are zero if $k=1$. Then $N\sim_{2}M$.

###### Proof.

This follows by unwinding the definitions and using the fact
$\lim_{\lambda\longrightarrow\infty}\frac{[\lambda]}{\lambda}=1.$ ∎

######  Proposition 7.6.

Let $\widetilde{M}:\mathbb{R}\longrightarrow\mathbb{R}$ be defined as

$\widetilde{M}(\gamma):=c_{1}\gamma^{d}+c_{2}\gamma^{d-1},$

where $d\in\mathbb{Z}^{\geq 1}$ and $c_{1},c_{2}$ are non-zero real numbers.
Let $\widetilde{N}:\mathbb{R}\longrightarrow\mathbb{R}$ be a function such
that $\widetilde{N}\sim_{2}\widetilde{M}$. Let us define
$\gamma:\mathbb{R}\longrightarrow\mathbb{R}$ as

$\gamma(\lambda):=a\lambda+b,$

where $a\in\mathbb{R}^{>0}$ and $b\in\mathbb{R}$. Define
$N,M:\mathbb{R}\longrightarrow\mathbb{R}$ as

$N(\lambda):=\widetilde{N}(\gamma(\lambda))\qquad\textnormal{and}\qquad
M(\lambda):=c_{1}a^{d}\lambda^{d}+(bdc_{1}+c_{2})a^{d-1}\lambda^{d-1}.$

If $(bdc_{1}+c_{2})\neq 0$, then $N\sim_{2}M$.

###### Proof.

The proof will follow by unwinding the definition of a limit. First of all, we
note that

$\displaystyle\frac{N(\lambda)}{\lambda^{d}}$
$\displaystyle=\Big{(}\frac{\widetilde{N}(\gamma(\lambda))}{\gamma(\lambda)^{d}}\Big{)}\Big{(}\frac{\gamma(\lambda)^{d}}{\lambda^{d}}\Big{)}.$
(57)

We now note that

$\displaystyle\lim_{\lambda\longrightarrow\infty}\Big{(}\frac{\widetilde{N}(\gamma(\lambda))}{\gamma(\lambda)^{d}}\Big{)}$
$\displaystyle=c_{1}\qquad\textnormal{and}$ (58)
$\displaystyle\lim_{\lambda\longrightarrow\infty}\Big{(}\frac{\gamma(\lambda)^{d}}{\lambda^{d}}\Big{)}$
$\displaystyle=a^{d}.$ (59)

Equation (58) follows from the fact that
$\lim_{\lambda\longrightarrow\infty}\frac{\widetilde{N}(\lambda)}{\lambda^{d}}=c_{1}$
and the fact that $\gamma(\lambda)$ is a monotonically increasing function of
$\lambda$ and applying the definition of a limit. Equations (58), (59) and
(57) imply that

$\displaystyle\lim_{\lambda\longrightarrow\infty}\frac{N(\lambda)}{\lambda^{d}}=c_{1}a^{d}.$
(60)

Next, we note that

$\displaystyle\frac{N(\lambda)-c_{1}a\lambda^{d}}{\lambda^{d-1}}$
$\displaystyle=\Big{(}\frac{\widetilde{N}(\gamma(\lambda))-c_{1}\gamma(\lambda)^{d}+c_{1}\gamma(\lambda)^{d}-c_{1}a^{d}\lambda^{d}}{\gamma(\lambda)^{d-1}}\Big{)}\Big{(}\frac{\gamma(\lambda)^{d-1}}{\lambda^{d-1}}\Big{)}$
$\displaystyle=\Big{(}\frac{\widetilde{N}(\gamma(\lambda))-c_{1}\gamma(\lambda)^{d}}{\gamma(\lambda)^{d-1}}\Big{)}\Big{(}\frac{\gamma(\lambda)^{d-1}}{\lambda^{d-1}}\Big{)}+\Big{(}\frac{c_{1}\gamma(\lambda)^{d}-c_{1}a^{d}\lambda^{d}}{\lambda^{d-1}}\Big{)}$
$\displaystyle\implies\lim_{\lambda\longrightarrow\infty}\Big{(}\frac{N(\lambda)-c_{1}a\lambda^{d}}{\lambda^{d-1}}\Big{)}$
$\displaystyle=\lim_{\lambda\longrightarrow\infty}\Big{(}\frac{\widetilde{N}(\gamma(\lambda))-c_{1}\gamma(\lambda)^{d}}{\gamma(\lambda)^{d-1}}\Big{)}a^{d-1}+\lim_{\lambda\longrightarrow\infty}\Big{(}\frac{c_{1}\gamma(\lambda)^{d}-c_{1}a^{d}\lambda^{d}}{\lambda^{d-1}}\Big{)}$
$\displaystyle=a^{d-1}c_{2}+a^{d-1}bdc_{1}.$ (61)

The first limit on the right hand side of equation (61) is computed using the
fact that $\gamma(\lambda)$ is a monotonically increasing function of
$\lambda$ and the fact that $\widetilde{N}\sim_{2}\widetilde{N}$ and then
applying definition of a limit. The second limit can be computed immediately
by considering the binomial expansion of $(a\lambda+b)^{d}$. Equations (60)
and (61) imply that $N\sim_{2}M$, whence the claim is proved. ∎

We are now ready to prove the main claim.

######  Theorem 7.7.

Let $d\in\mathbb{Z}^{\geq 1}$, $y\in\mathbb{R}$ and $A\in\mathbb{R}^{>0}$.
Define the functions $N,M:\mathbb{R}\longrightarrow\mathbb{R}$ as

$\displaystyle N(\lambda)$
$\displaystyle:=\sum_{\begin{subarray}{c}m-y\in\mathbb{Z}^{\geq 0}\\\
Am^{2}\leq\lambda^{2}\end{subarray}}m^{d-1}\qquad\textnormal{and}\qquad
M(\lambda):=\frac{1}{d(\sqrt{A})^{d}}\lambda^{d}+\frac{1}{2(\sqrt{A})^{d-1}}\lambda^{d-1}.$

Then $N\sim_{2}M$.

###### Proof.

Let us denote $\tilde{m}:=m-y$. Then $\tilde{m}\in\mathbb{Z}^{\geq 0}$ and the
condition
$Am^{2}\leq\lambda^{2}\Leftrightarrow-\frac{\lambda}{\sqrt{A}}-y\leq\tilde{m}\leq\frac{\lambda}{\sqrt{A}}-y$,
owing to the fact $\tilde{m}\in\mathbb{Z}^{\geq 0}$, becomes
$0\leq\tilde{m}\leq\left[\frac{\lambda}{\sqrt{A}}-y\right]$.
Let $\gamma(\lambda):=\frac{\lambda}{\sqrt{A}}-y$. Then

$\displaystyle N(\lambda)$
$\displaystyle=\sum_{\tilde{m}=0}^{[\gamma(\lambda)]}(\tilde{m}+y)^{d-1}$ (62)
$\displaystyle=\sum_{\tilde{m}=0}^{[\gamma(\lambda)]}\left(\tilde{m}^{d-1}+(d-1)\tilde{m}^{d-2}y+\cdots+y^{d-1}\right).$

Using corollary 7.2, we note that

$\sum_{\tilde{m}=0}^{[\gamma(\lambda)]}\tilde{m}^{d-1}=\frac{1}{d}[\gamma(\lambda)]^{d}+\frac{1}{2}[\gamma(\lambda)]^{d-1}+\text{terms
with lower powers of $[\gamma(\lambda)]$},$

and

$\sum_{\tilde{m}=0}^{[\gamma(\lambda)]}(d-1)\tilde{m}^{d-2}y=y[\gamma(\lambda)]^{d-1}+\frac{1}{2}(d-1)y[\gamma(\lambda)]^{d-2}+\text{terms
with lower powers of $[\gamma(\lambda)]$}.$

Using this in (62), we have

$\displaystyle N(\lambda)$
$\displaystyle=\widetilde{N}(\gamma(\lambda)),\qquad\textnormal{where}$
$\displaystyle\widetilde{N}(\gamma)$
$\displaystyle=\frac{1}{d}[\gamma]^{d}+\left(\frac{1}{2}+y\right)[\gamma]^{d-1}+\text{terms
with lower powers of $[\gamma]$}.$

Let us now define $\widetilde{M}:\mathbb{R}\longrightarrow\mathbb{R}$ as

$\displaystyle\widetilde{M}(\gamma)$
$\displaystyle:=\frac{1}{d}\gamma^{d}+\left(\frac{1}{2}+y\right)\gamma^{d-1}.$

By proposition 7.5, $\widetilde{N}\sim_{2}\widetilde{M}$. As a result, using
proposition 7.6, we conclude that $N\sim_{2}M$, whence the claim is proved. ∎

######  Remark 7.8.

Note that in the expression for $M(\lambda)$, the coefficient of
$\lambda^{d-1}$ is non-zero. Hence, theorem 7.7 shows that the error term for
$N(\lambda)$ can not be improved beyond $\lambda^{d-1}$ in addition to being
no worse than $\lambda^{d-1}$.

Using the above theorem, we next prove the following result.

######  Theorem 7.9.

Let $d\in\mathbb{Z}^{\geq 2}$ and $\mathsf{Q}_{d-1}(t)$ be a degree $d-1$
polynomial in $t$ that does not have the $t^{d-2}$ term. In other words,

$\displaystyle\mathsf{Q}_{d-1}(t)$
$\displaystyle=Ct^{d-1}+\varphi(t),\qquad\textnormal{if}\qquad
d>2\qquad\textnormal{and}$ $\displaystyle\mathsf{Q}_{d-1}(t)$
$\displaystyle=Ct,\qquad\textnormal{if}\qquad d=2,$

where $\varphi(t)$ is a polynomial of degree at most $d-3$ and $C$ is some
non-zero constant. Let $y$ be a fixed real number and $A$ a fixed positive
real numbers. Define $N,M:\mathbb{R}\longrightarrow\mathbb{R}$ as

$\displaystyle N(\lambda)$
$\displaystyle:=\sum_{\begin{subarray}{c}m-y\in\mathbb{Z}^{\geq 0}\\\
Am^{2}\leq
a\lambda^{2}+b\end{subarray}}\mathsf{Q}_{d-1}(m)\qquad\textnormal{and}$
$\displaystyle M(\lambda)$
$\displaystyle:=\frac{Ca^{\frac{d}{2}}}{d\left(\sqrt{A}\right)^{d}}\lambda^{d}+\frac{Ca^{\frac{d-1}{2}}}{2\left(\sqrt{A}\right)^{d-1}}\lambda^{d-1},$

where $a\in\mathbb{R}^{>0}$ and $b\in\mathbb{R}$. Then $N\sim_{2}M$.

###### Proof.

This follows from theorem 7.7. By applying binomial theorem to
$\Big{(}\frac{a\lambda^{2}+b}{A}\Big{)}^{\frac{d}{2}}$, the leading term gives
us the desired asymptotics. The lower order terms can not cancel off or change
the error term, because the $t^{d-2}$ term is absent in $\mathsf{Q}_{d-1}(t)$.
∎∎

Proof of theorem 3.3: Follows immediately from theorem 7.9. ∎

##  8\. Acknowledgement

We are grateful to Antareep Saud for several useful discussions related to
this project. The third author is also grateful to Vamsi Pingali for helpful
discussions and comments and in particular pointing out the reference [2] for
the proof of Weyl’s law.

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