LXVIII.4 (1994)
Mean square limit for lattice points in a sphere
by
Pavel M. Bleher (Indianapolis, Ind.) and Freeman J. Dyson (Princeton, N.J.) 1. Main result. Let
N (R) = #{n ∈ Z3: |n| ≤ R}
be the number of integral points inside a sphere of radius R centered at the origin and
F (R) = R−1(N (R) − (4/3)πR3).
We prove in this note the following result:
Theorem 1.1.
(1.1) lim
T →∞(T log T )−1
T
R
1
|F (R)|2dR = K with K = (32/7) (ζ(2)/ζ(3)).
The three-dimensional case is the most difficult one. A version of The- orem 1.1 is known for a long time for the circle (see [Cra] and [Lan1]) and for the d-dimensional ball when d ≥ 4 (see [Wal]). In [Ble1] a similar state- ment was proved for any strictly convex (in the sense that the curvature of the boundary is positive everywhere) oval in the plane with the origin inside the oval. Making an analogy to the theory of renormalization group in statistical mechanics we may say that three is the critical dimension for the problem under consideration. Namely, the series of squared Fourier am- plitudes of N (R) converges when d < 3 and diverges when d ≥ 3 (in fact logarithmically diverges when d = 3). The criticality of d = 3 is reflected then in the appearance of the log-correction in (1.1).
It is easy to show that
T →∞lim T−1
T
R
1
F (R) dR = 0,
[383]
hence Theorem 1.1 implies that
F (R) = Ω±(log1/2R)
(which means that lim supR→∞(log R)−1/2(±F (R)) > 0). The estimate F (R) = Ω−(log1/2R) was proved long ago by Szeg¨o [Sze], and Nowak [Now]
proved the estimate
F (R) = Ω−(log1/3R)
for any strictly convex domain in R3 with the origin inside the domain.
The estimate F (R) = Ω+(log1/2R) is new and it improves the recent re- sult F (R) = Ω+(log log R) by Adhikari and P´etermann [AP] (the authors thank Andrzej Schinzel for calling their attention to the works of Szeg¨o and Adhikari and P´etermann).
See also earlier works [Wal], [CN] and [BK] where somewhat weaker estimates were obtained.
As concerns O-results, Landau [Lan2] proved that F (R) = O(Rθ)
with θ = 1/2. Vinogradov [Vin1] strengthened this result to θ = 5/14 + ε,
∀ε > 0, and Chen Jing-Run [Che] and Vinogradov [Vin2] to θ = 1/3 + ε.
Recently Chamizo and Iwaniec [CI] strengthened this further to θ = 7/22+ε.
Randol [Ran] proved the O-estimate with θ = 1/2 for any strictly convex domain in R3 (for ellipsoids it was known before from the work of Landau [Lan2]). Recently Kr¨atzel and Nowak [KN] improved the O-estimate of Randol to θ = 8/17 + ε. For a review and many other results on counting lattice points in multidimensional spheres see [Wal] and [Gro].
Theorem 1.1 can be reformulated in a probability language, namely, that the variance of the random variable (log R)−1/2F (R), assuming that R is uniformly distributed on [1, T ], converges to K > 0 as T → ∞. An interesting and natural question is then: what is the limiting distribution of (log R)−1/2F (R) (if the latter exists)? We have no answer to this question, but the following simple heuristic argument speaks in favor of a Gaussian limiting distribution.
By formulas (1.12), (1.13) below, (1.2) F (R) = −π−1 X
n∈Z3\{0}
ϕ(2π|n|δ)|n|−2cos(2π|n|R) + ε(R), where ϕ(x) is a C∞function fast decreasing at ∞ and ε(R) is an error term.
This represents F (R) as an almost periodic function plus an error. We can group terms with commensurate frequencies in (1.2) and rewrite (1.2) as
(1.3) F (R) = X
square free k
fk(k1/2R) + ε(R),
where
fk(t) = −π−1k−1
∞
X
l=1
l−2r3(l2k)ϕ(2πlk1/2δ) cos(2πlt)
are bounded periodic functions with period 1 and r3(k) = #{n ∈ Z3 :
|n|2= k}. The numbers k1/2with square free k (i.e., k 6= l2k0with l > 1) are linearly independent over Z and so the random variables {{k1/2R}, k square free} are asymptotically independent in the limit T → ∞. Therefore for a fixed δ > 0, the limiting distribution of the sum in (1.3) is the same as the limiting distribution of the random series
ξ = X
square free k
fk(θk),
where {θk} are independent random variables uniformly distributed on [0, 1].
Observe that
Var fk(θk) =
1
R
0
fk(t)2dt
is of order of k−2(r3(k))2(ϕ(2πk1/2δ))2. The series P k−2(r3(k))2 is loga- rithmically divergent (see Section 2 below), hence Var ξ ∼ C|log δ|−1. Thus ξ is a series of uniformly bounded independent random variables and the variance of ξ diverges as δ → 0. By the Lindeberg theorem this implies that (Var ξ)−1/2ξ converges to a standard Gaussian distribution, so that the limiting distribution of (Var F (R))−1/2F (R) is standard Gaussian as well.
The weakness of this argument is that we took the limit T → ∞ first and the limit δ → 0 second, while in (1.2) δ = T−1λ(T ) with (log T )−1/2λ(T ) → 0, so that δ → 0 simultaneously with T → ∞. This explains why the argument is only heuristic.
For the circle problem Heath-Brown [H-B] and Bleher, Cheng, Dyson and Lebowitz [BCDL] proved that the limiting distribution of (Var F (R))−1/2
× F (R) exists and is non-Gaussian with an analytic density decreasing at infinity roughly as exp(−cx4).
P r o o f o f T h e o r e m 1.1. Since
(1.4) N (R) = X
n∈Z3
χ(n; R),
where χ(x; R) is the characteristic function of the ball {|x| ≤ R}, the Poisson summation formula implies
N (R) − (4/3)πR3=X0
n
χ(2πn; R),e
whereP0 n=P
n∈Z3\{0}and χ(ξ; R) is the Fourier transform of χ(x; R),e χ(ξ; R) =e R
|x|≤R
eixξdx = R3 R
|x|≤1
eixRξdx.
Since in addition,
R
|x|≤1
eixηdx = π
1
R
−1
cos(|η|r)(1 − r2) dr = 4π|η|−3(sin |η| − |η| cos |η|), we obtain
(1.5) N (R) − (4/3)πR3= R3X0 n
J (2π|n|R), with
(1.6) J (t) = 4πt−3(sin t − t cos t).
The series in (1.5) is only conditionally convergent. Define for δ > 0,
(1.7) Nδ(R) = X
n∈Z3
χδ(n; R) with
(1.8) χδ(x; R) = δ−3 R
|y|≤R
ϕ(δe −1(x − y)) dy,
whereϕ(x) ≥ 0 is a Ce ∞ isotropic (ϕ(x) =e ϕe0(|x|)) cap with ϕ(x) = 0 whene
|x| ≥ 1 and R
R3ϕ(x) dx = 1. Then again by Poisson’s summation,e (1.9) Nδ(R) − (4/3)πR3= R3X0
n
ϕ(2πnδ)J (2π|n|R) with a convergent series on the right. Put
(1.10) δ = T−1λ(T ),
where λ(T ) is a slowly increasing function with
(1.11) lim
T →∞λ(T ) = ∞ and lim
T →∞(log T )−1/2λ(T ) = 0.
From (1.9) and (1.6), (1.12) Fδ(R)
:= R−1(Nδ(R) − (4/3)πR3)
= − π−1X0 n
ϕ(2πnδ)|n|−2cos(2π|n|R) + O(R−1Tε), ∀ε > 0.
We will prove the following two lemmas, from which Theorem 1.1 follows.
Lemma 1.2.
(1.13) lim
T →∞(T log T )−1
T
R
1
|F (R) − Fδ(R)|2dR = 0.
Lemma 1.3.
(1.14) lim
T →∞(T log T )−1
T
R
1
|Fδ(R)|2dR = K > 0.
P r o o f o f L e m m a 1.2. We follow the proof of Lemma 4.3 in [Ble2].
To simplify notations we denote by c and c0various constants which can be different in different estimates. We have
Fδ(R) − F (R) = R−1X
n
(χδ(n; R) − χ(n; R)), so
I ≡ (T log T )−1
T
R
1
|Fδ(R) − F (R)|2dR =X
m,n
I(m, n) with
I(m, n) = (T log T )−1
T
R
1
(χδ(m; R) − χ(m; R))(χδ(n; R) − χ(n; R))R−2dR.
Observe that I(m, n) = 0 unless ||m| − |n|| < 2δ and |m|, |n| < T + δ. In addition, I(m, n) ≤ c(T log T )−1|n|−2δ, for all m, n, hence
(1.15) I ≤ c(T log T )−1δ X
n:|n|≤T +δ
|n|−2 X
m:||m|−|n||≤2δ
1.
Let us estimate
I0= X
n:T /2≤|n|≤T
X
m:||m|−|n||≤2δ
1.
Observe that ||m|2− |n|2| ≤ ||m| + |n||2δ ≤ 5T δ = 5λ(T ), so if |n| is fixed we have not more than 10λ(T ) possibilities for |m|. Therefore
I0≤ X
T /2≤√ k≤T
r3(k) X
|k−l|≤5λ(T )
r3(l),
where r3(k) = #{n ∈ Z3: |n|2= k}, and I0≤ cλ(T ) X
k≤(2T )2
r3(k)2 (use r3(k)r3(l) ≤ r3(k)2+ r3(l)2). Since
(1.16) lim
N →∞N−2 X
k≤N
r3(k)2= K0= 16
7 π2ζ(2) ζ(3)
(see the next section) we obtain
I0≤ cλ(T )T4.
Applying this estimate for T := 2T, T, T /2, T /4, . . . , we obtain (by (1.15)) (1.17) I ≤ cλ(T )2(log T )−1,
and Lemma 1.2 follows (use the second limit in (1.11)).
P r o o f o f L e m m a 1.3. Let I = (T log T )−1
T
R
1
X0 n
ϕ(2πnδ)|n|−2cos(2π|n|R)
2
dR.
We will prove that
T →∞lim I = K1= 32 7 π2ζ(2)
ζ(3). Then by (1.12), Lemma 1.3 will follow with K = π−2K1.
Since ϕ(ξ) is isotropic, ϕ(ξ) = ϕ0(|ξ|), I = (T log T )−1
T
R
1
∞
X
k=1
r3(k)k−1ϕ0(2π
√
kδ) cos(2π
√ kR)
2
dR.
Hence
I =
∞
X
k,l=1
r3(k)r3(l)k−1l−1ϕ0(2π
√
kδ)ϕ0(2π
√
lδ)A(k, l) with
A(k, l) = (T log T )−1
T
R
1
cos(2π
√
kR) cos(2π
√
lR) dR.
Since
A(k, k) = (1/2)(log T )−1(1 + O(T−1)), T → ∞, the diagonal contribution to I is
Idiag=
∞
X
k=1
r3(k)2k−2ϕ0(2π
√
kδ)2((1/2)(log T )−1+ O((T log T )−1)).
Since ϕ0(2π√
kδ) produces a smooth cutoff at the scale δ−2 and (1.18)
N
X
k=1
r3(k)2k−2= K1(log N )(1 + o(1)), N → ∞, (see the next section), we obtain
Idiag= K1(log T2)(1/2)(log T )−1(1 + o(1)) (1.19)
= K1(1 + o(1)), T → ∞.
For k 6= l,
|A(k, l)| ≤ c(T log T )−1|√ k −√
l|−1, so the off-diagonal contribution to I is estimated as
|Ioff| ≤X
k6=l
r3(k)r3(l)k−1l−1|ϕ0(2π
√
kδ)ϕ0(2π
√
lδ)|(T log T )−1|√ k −
√ l|−1. By symmetry we may consider only l > k. For any p ≥ 1, consider the block Ioff(p) in Ioff with 2p ≤ k ≤ 2p+1 and 0 < j ≡ l − k ≤ 2p. Since
|ϕ0(2π√
kδ)| ≤ c(1 + kδ2)−3, this block is estimated as
|Ioff(p)| ≤ c(1 + 2pδ2)−32−2p
2p+1
X
k=2p 2p
X
j=1
r3(k)r3(k + j)(T log T )−12p/2j−1 (use √
k + j −√
k = j(√
k + j +√
k)−1≥ cj2−p/2). Now, r3(k)r3(k + j) ≤ r3(k)2+ r3(k + j)2 and by (1.16),
2p+1
X
k=2p
r3(k + j)2≤ c22p, hence
|Ioff(p)| ≤ c(1 + 2pδ2)−3(T log T )−12p/2
2p
X
j=1
j−1 (1.20)
≤ c0(1 + 2pδ2)−3(T log T )−12p/2p.
Consider now the block Ioff(p, q) in Ioff with 2p≤ k ≤ 2p+1 and 2p+q ≤ j ≡ l − k ≤ 2p+q+1, where p ≥ 1 and q ≥ 0. This block is estimated as
|Ioff(p, q)| ≤ c(1 + 2pδ2)−32−p2−2p−2q(T log T )−12p/2
×
2p+1
X
k=2p 2p+q+1
X
j=2p+q
r3(k)r3(k + j).
Since
X
k≤N
r3(k) = 4π
3 N3/2(1 + o(1)), we have
2p+q+1
X
j=2p+q
r3(k + j) ≤ c2(3/2)(p+q) and
2p+1
X
k=2p
r3(k) ≤ c2(3/2)p, and thus
(1.21) |Ioff(p, q)| ≤ c(1 + 2pδ2)−32−q/2(T log T )−12p/2.
Combining this estimate with (1.20) we obtain
|Ioff(p)| +
∞
X
q=0
|Ioff(p, q)| ≤ c(1 + 2pδ2)−3(T log T )−12p/2p,
and summing now in p = 1, 2, . . . we arrive at |Ioff| ≤ c (T log T )−1δ−1|log δ|.
Since δ = T−1λ(T ), this implies |Ioff| ≤ cλ(T )−1. Hence limT →∞Ioff = 0 and
T →∞lim I = lim
T →∞Idiag= K1
(use (1.19)), which proves Lemma 1.3.
2. Evaluation of N−2PN
k=1|r3(k)|2. Let θ(z) =P∞
n=−∞zn2. Then S ≡ (2πi)−1 R
|z|=e−2πδ
θ3(z)θ3(z)(dz/z) =
∞
X
k=0
e−4πkδ|r3(k)|2.
Assuming δ → 0 we use the singular series of Hardy (see, e.g., [Gro] or [Vau]). We have (see, e.g., [Gro, p. 151])
θ(e2πih/k−2πz) = (k√
2z)−1G(h, k) + . . . , hence
|θ(e2πih/k−2πz)|6= (8k6|z|3)−1|G(h, k)|6+ . . . , where G(h, k) =Pk−1
j=0e2πihj2/k is the Gaussian sum. Now, (2π)−1 R
|θ(e2πih/k−2π(δ−i(ξ/2π)))|6dξ
= (8k6)−1|G(h, k)|6 R
|δ − iξ|−3dξ + . . .
= (8k6)−1|G(h, k)|6δ−2
∞
R
−∞
(x2+ 1)−3/2dx + . . .
= δ−2(4k6)−1|G(h, k)|6+ . . . , hence
S = (2δ)−2X
h,k
k−6|G(h, k)|6+ . . . , so that
(2.1) S = (2δ)−2
∞
X
k=1
Ak+ . . .
with
Ak = k−6 X
h mod k;(h,k)=1
|G(h, k)|6. The dots in (2.1) stand for o(δ−2), δ → 0.
Ak is a multiplicative arithmetical function (see [Gro, p. 156]), so that Ak1k2= Ak1Ak2 when (k1, k2) = 1. Hence
(2.2)
∞
X
k=1
Ak=Y
p
(1 + Ap+ Ap2+ . . .)
with the product over primes. Now, A2 = 0 and for a > 1, |G(h, 2a)| = 2(a+1)/2 (see [Gro, p. 138]), hence A2a = 2−6a23(a+1)2a−1= 2−2a+2, a > 1, and
1 + A2+ A4+ A8+ . . . = 1 + 1/3 = 4/3.
When p > 2, |G(h, pa)| = |(h/pa)| |G(1, pa)| = pa/2 (see [Gro, p. 138]), hence Apa = p−6ap3aP
h1 = (p − 1)p−2a−1 and
1 + Ap+ Ap2+ . . . = 1 + (p(p + 1))−1. Therefore from (2.1) and (2.2),
S = (2δ)−24 3
Y
p>2
{1 + (p(p + 1))−1} + . . . = δ−22 7
Y
p
{1 + (p(p + 1))−1} + . . . Now, Q
p{1 + (p(p + 1))−1} = ζ(2)/ζ(3), hence
(2.3) S = δ−22
7 ζ(2) ζ(3) + . . . Thus
δ→0limδ2
∞
X
k=1
|r3(k)|2exp(−4πkδ) = 2 7
ζ(2) ζ(3) and so
δ→0limδ2
∞
X
k=1
|r3(k)|2exp(−kδ) = 32 7 π2ζ(2)
ζ(3).
By the tauberian theorem of Hardy and Littlewood [HL] this implies that
N →∞lim N−2
N
X
k=1
|r3(k)|2= 16
7 π2ζ(2) ζ(3). Hence by partial summation,
N →∞lim (log N )−1
N
X
k=1
|r3(k)|2 k2 = 32
7 π2ζ(2) ζ(3).
Acknowledgements. The authors thank Fernando Chamizo and Hen- ryk Iwaniec for useful remarks and some references. The work is supported by a grant from the Ambrose Monell Foundation and by a grant in aid
# DE-FG02-90ER40542 from the US Department of Energy.
References
[AP] S. D. A d h i k a r i and Y.-F. S. P ´e t e r m a n n, Lattice points in ellipsoids, Acta Arith. 59 (1991), 329–338.
[Ble1] P. M. B l e h e r, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. J. 67 (1992), 461–481.
[Ble2] —, Distribution of energy levels of a quantum free particle on a surface of revolution, ibid. 74 (1994), 45–93.
[BCDL] P. M. B l e h e r, Z. C h e n g, F. J. D y s o n and J. L. L e b o w i t z, Distribution of the error term for the number of lattice points inside a shifted circle, Comm.
Math. Phys. 154 (1993), 433–469.
[BK] M. N. B l e i c h e r and M. I. K n o p p, Lattice points in a sphere, Acta Arith. 10 (1965), 369–376.
[CI] F. C h a m i z o and H. I w a n i e c, A 3-dimensional lattice point problem, in prepa- ration.
[CN] K. C h a n d r a s e k h a r a n and R. N a r a s i m h a n, Hecke’s functional equation and the average order of arithmetical functions, Acta Arith. 6 (1961), 487–503.
[Che] J.-R. C h e n, Improvement on the asymptotic formulas for the number of lattice points in a region of the three dimensions (II ), Sci. Sinica 12 (1963), 751–764.
[Cra] H. C r a m ´e r, ¨Uber zwei S¨atze von Herrn G. H. Hardy , Math. Z. 15 (1922), 201–210.
[Gro] E. G r o s s w a l d, Representations of Integers as Sums of Squares, Springer, New York, 1985.
[HL] G. H. H a r d y and J. E. L i t t l e w o o d, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, Proc. London Math.
Soc. 13 (1914), 174.
[H-B] D. R. H e a t h-B r o w n, The distribution and moments of the error term in the Dirichlet divisor problem, Acta Arith. 60 (1992), 389–415.
[KN] E. K r ¨a t z e l and W. G. N o w a k, Lattice points in large convex bodies, II , ibid.
62 (1992), 285–295.
[Lan1] E. L a n d a u, Vorlesungen ¨uber Zahlentheorie, V. 1, Hirzel, Leipzig, 1927.
[Lan2] —, Ausgew¨ahlte Abhandlungen zur Gitterpunktlehre, A. Walfisz (ed.), Deutsch- er Verlag der Wiss., Berlin, 1962.
[Now] W. G. N o w a k, On the lattice rest of a convex body in Rs, II , Arch. Math.
(Basel) 47 (1986), 232–237.
[Ran] B. R a n d o l, A lattice point problem, I, II , Trans. Amer. Math. Soc. 121 (1966), 257–268; 125 (1966), 101–113.
[Sze] G. S z e g ¨o, Beitr¨age zur Theorie der Laguerreschen Polynome. II : Zahlentheo- retische Anwendungen, Math. Z. 25 (1926), 388–404.
[Vau] R. C. V a u g h a n, The Hardy–Littlewood Method , Cambridge University Press, Cambridge, 1981.
[Vin1] I. M. V i n o g r a d o v, On the number of integral points in a given domain, Izv.
Akad. Nauk SSSR Ser. Mat. 24 (1960), 777–786.
[Vin2] I. M. V i n o g r a d o v, On the number of integral points in a sphere, ibid. 27 (1963), 957–968.
[Wal] A. W a l f i s z, Gitterpunkte in mehrdimensionalen Kugeln, PWN, Warszawa, 1957.
DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF NATURAL SCIENCES
INDIANA UNIVERSITY– INSTITUTE FOR ADVANCED STUDY
PURDUE UNIVERSITY INDIANAPOLIS PRINCETON, NEW JERSEY 08540
402 BLACKFORD STREET U.S.A.
INDIANAPOLIS, INDIANA 46202-3272 U.S.A.
E-mail: BLEHER@MATH.IUPUI.EDU
Received on 12.3.1994
and in revised form on 25.4.1994 (2584)
