Mean square limit for lattice points in a sphere

Transcription

1 ACTA ARITHMETICA LXVIII.4 (994) Mean square limit for lattice points in a sphere by Pavel M. Bleher (Indianapolis, Ind.) and Freeman J. Dyson (Princeton, N.J.). Main result. Let N(R) = #{n Z 3 : n R} be the number of integral points inside a sphere of radius R centered at the origin and F (R) = R (N(R) (4/3)πR 3 ). We prove in this note the following result: Theorem.. (.) lim (T log T ) T with K = (32/7) (ζ(2)/ζ(3)). T F (R) 2 dr = K The three-dimensional case is the most difficult one. A version of Theorem. is known for a long time for the circle (see [Cra] and [Lan]) and for the d-dimensional ball when d 4 (see [Wal]). In [Ble] a similar statement 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 amplitudes 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 (.). It is easy to show that lim T T T F (R) dr = 0, [383]

2 384 P. M. Bleher and F. J. Dyson hence Theorem. implies that F (R) = Ω ± (log /2 R) (which means that lim sup R (log R) /2 (±F (R)) > 0). The estimate F (R) = Ω (log /2 R) was proved long ago by Szegö [Sze], and Nowak [Now] proved the estimate F (R) = Ω (log /3 R) for any strictly convex domain in R 3 with the origin inside the domain. The estimate F (R) = Ω + (log /2 R) is new and it improves the recent result F (R) = Ω + (log log R) by Adhikari and Pétermann [AP] (the authors thank Andrzej Schinzel for calling their attention to the works of Szegö and Adhikari and Pétermann). 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(R θ ) with θ = /2. Vinogradov [Vin] strengthened this result to θ = 5/4 + ε, ε > 0, and Chen Jing-Run [Che] and Vinogradov [Vin2] to θ = /3 + ε. Recently Chamizo and Iwaniec [CI] strengthened this further to θ = 7/22+ε. Randol [Ran] proved the O-estimate with θ = /2 for any strictly convex domain in R 3 (for ellipsoids it was known before from the work of Landau [Lan2]). Recently Krätzel and Nowak [KN] improved the O-estimate of Randol to θ = 8/7 + ε. For a review and many other results on counting lattice points in multidimensional spheres see [Wal] and [Gro]. Theorem. can be reformulated in a probability language, namely, that the variance of the random variable (log R) /2 F (R), assuming that R is uniformly distributed on [, T ], converges to K > 0 as T. An interesting and natural question is then: what is the limiting distribution of (log R) /2 F (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 (.2), (.3) below, (.2) F (R) = π ϕ(2π n δ) n 2 cos(2π n R) + ε(r), n Z 3 \{0} 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 (.2) and rewrite (.2) as (.3) F (R) = f k (k /2 R) + ε(r), square free k

3 where f k (t) = π k Lattice points in a sphere 385 l= l 2 r 3 (l 2 k)ϕ(2πlk /2 δ) cos(2πlt) are bounded periodic functions with period and r 3 (k) = #{n Z 3 : n 2 = k}. The numbers k /2 with square free k (i.e., k l 2 k with l > ) are linearly independent over Z and so the random variables {{k /2 R}, k square free} are asymptotically independent in the limit T. Therefore for a fixed δ > 0, the limiting distribution of the sum in (.3) is the same as the limiting distribution of the random series ξ = f k (θ k ), square free k where {θ k } are independent random variables uniformly distributed on [0, ]. Observe that Var f k (θ k ) = 0 f k (t) 2 dt is of order of k 2 (r 3 (k)) 2 (ϕ(2πk /2 δ)) 2. The series k 2 (r 3 (k)) 2 is logarithmically divergent (see Section 2 below), hence Var ξ C log δ. 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) /2 ξ converges to a standard Gaussian distribution, so that the limiting distribution of (Var F (R)) /2 F (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 (.2) δ = T λ(t ) with (log T ) /2 λ(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)) /2 F (R) exists and is non-gaussian with an analytic density decreasing at infinity roughly as exp( cx 4 ). P r o o f o f T h e o r e m.. Since (.4) N(R) = n Z 3 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)πR 3 = χ(2πn; R), n

4 386 P. M. Bleher and F. J. Dyson where n = n Z 3 \{0} and χ(ξ; R) is the Fourier transform of χ(x; R), χ(ξ; R) = Since in addition, e ixη dx = π x e ixξ dx = R 3 x R x e ixrξ dx. cos( η r)( r 2 ) dr = 4π η 3 (sin η η cos η ), we obtain (.5) N(R) (4/3)πR 3 = R 3 J(2π n R), with (.6) J(t) = 4πt 3 (sin t t cos t). The series in (.5) is only conditionally convergent. Define for δ > 0, (.7) N δ (R) = χ δ (n; R) n Z 3 n with (.8) χ δ (x; R) = δ 3 y R ϕ(δ (x y)) dy, where ϕ(x) 0 is a C isotropic ( ϕ(x) = ϕ 0 ( x )) cap with ϕ(x) = 0 when x and ϕ(x) dx =. Then again by Poisson s summation, R 3 (.9) N δ (R) (4/3)πR 3 = R 3 ϕ(2πnδ)j(2π n R) with a convergent series on the right. Put (.0) δ = T λ(t ), where λ(t ) is a slowly increasing function with (.) lim λ(t ) = and lim (log T T T ) /2 λ(t ) = 0. From (.9) and (.6), (.2) F δ (R) := R (N δ (R) (4/3)πR 3 ) n = π ϕ(2πnδ) n 2 cos(2π n R) + O(R T ε ), ε > 0. n We will prove the following two lemmas, from which Theorem. follows.

5 Lemma.2. (.3) lim (T log T ) T Lemma.3. (.4) lim (T log T ) T Lattice points in a sphere 387 T T F (R) F δ (R) 2 dr = 0. F δ (R) 2 dr = K > 0. P r o o f o f L e m m a.2. We follow the proof of Lemma 4.3 in [Ble2]. To simplify notations we denote by c and c 0 various constants which can be different in different estimates. We have so F δ (R) F (R) = R n I (T log T ) T (χ δ (n; R) χ(n; R)), F δ (R) F (R) 2 dr = m,n I(m, n) with I(m, n) = (T log T ) T (χ δ (m; R) χ(m; R))(χ δ (n; R) χ(n; R))R 2 dr. Observe that I(m, n) = 0 unless m n < 2δ and m, n < T + δ. In addition, I(m, n) c(t log T ) n 2 δ, for all m, n, hence (.5) I c(t log T ) δ n 2. Let us estimate I 0 = n: n T +δ n:t/2 n T m: m n 2δ m: m n 2δ Observe that m 2 n 2 m + n 2δ 5T δ = 5λ(T ), so if n is fixed we have not more than 0λ(T ) possibilities for m. Therefore I 0 r 3 (k) r 3 (l), T/2 k T k l 5λ(T ) where r 3 (k) = #{n Z 3 : n 2 = k}, and I 0 cλ(t ) r 3 (k) 2 k (2T ) 2 (use r 3 (k)r 3 (l) r 3 (k) 2 + r 3 (l) 2 ). Since (.6) lim N 2 r 3 (k) 2 = K 0 = 6 ζ(2) π2 N 7 ζ(3) k N.

6 388 P. M. Bleher and F. J. Dyson (see the next section) we obtain I 0 cλ(t )T 4. Applying this estimate for T := 2T, T, T/2, T/4,..., we obtain (by (.5)) (.7) I cλ(t ) 2 (log T ), and Lemma.2 follows (use the second limit in (.)). P r o o f o f L e m m a.3. Let T I = (T log T ) ϕ(2πnδ) n 2 cos(2π n R) 2 dr. We will prove that n lim I = K = 32 ζ(2) π2 T 7 ζ(3). Then by (.2), Lemma.3 will follow with K = π 2 K. Since ϕ(ξ) is isotropic, ϕ(ξ) = ϕ 0 ( ξ ), T I = (T log T ) r 3 (k)k ϕ 0 (2π kδ) cos(2π kr) 2 dr. Hence with I = k,l= k= r 3 (k)r 3 (l)k l ϕ 0 (2π kδ)ϕ 0 (2π lδ)a(k, l) A(k, l) = (T log T ) T cos(2π kr) cos(2π lr) dr. Since A(k, k) = (/2)(log T ) ( + O(T )), T, the diagonal contribution to I is I diag = r 3 (k) 2 k 2 ϕ 0 (2π kδ) 2 ((/2)(log T ) + O((T log T ) )). k= Since ϕ 0 (2π kδ) produces a smooth cutoff at the scale δ 2 and N (.8) r 3 (k) 2 k 2 = K (log N)( + o()), N, k= (see the next section), we obtain (.9) I diag = K (log T 2 )(/2)(log T ) ( + o()) = K ( + o()), T.

7 Lattice points in a sphere 389 For k l, A(k, l) c(t log T ) k l, so the off-diagonal contribution to I is estimated as I off k l r 3 (k)r 3 (l)k l ϕ 0 (2π kδ)ϕ 0 (2π lδ) (T log T ) k l. By symmetry we may consider only l > k. For any p, consider the block I off (p) in I off with 2 p k 2 p+ and 0 < j l k 2 p. Since ϕ 0 (2π kδ) c( + kδ 2 ) 3, this block is estimated as p+ 2 I off (p) c( + 2 p δ 2 ) 3 2 2p 2 p k=2 p j= r 3 (k)r 3 (k + j)(t log T ) 2 p/2 j (use k + j k = j( k + j + k) cj2 p/2 ). Now, and by (.6), hence (.20) r 3 (k)r 3 (k + j) r 3 (k) 2 + r 3 (k + j) 2 2 p+ k=2 p r 3 (k + j) 2 c2 2p, p 2 I off (p) c( + 2 p δ 2 ) 3 (T log T ) 2 p/2 j= c 0 ( + 2 p δ 2 ) 3 (T log T ) 2 p/2 p. Consider now the block I off (p, q) in I off with 2 p k 2 p+ and 2 p+q j l k 2 p+q+, where p and q 0. This block is estimated as Since we have and thus 2 p+q+ j I off (p, q) c( + 2 p δ 2 ) 3 2 p 2 2p 2q (T log T ) 2 p/2 2 p+ k=2 p 2 p+q+ j=2 p+q r 3 (k)r 3 (k + j). r 3 (k) = 4π 3 N 3/2 ( + o()), k N j=2 p+q r 3 (k + j) c2 (3/2)(p+q) and 2 p+ k=2 p r 3 (k) c2 (3/2)p, (.2) I off (p, q) c( + 2 p δ 2 ) 3 2 q/2 (T log T ) 2 p/2.

8 390 P. M. Bleher and F. J. Dyson Combining this estimate with (.20) we obtain I off (p) + I off (p, q) c( + 2 p δ 2 ) 3 (T log T ) 2 p/2 p, q=0 and summing now in p =, 2,... we arrive at I off c (T log T ) δ log δ. Since δ = T λ(t ), this implies I off cλ(t ). Hence lim T I off = 0 and lim I = lim I diag = K T T (use (.9)), which proves Lemma Evaluation of N 2 N k= r 3(k) 2. Let θ(z) = n= zn2. Then S (2πi) z =e 2πδ θ 3 (z)θ 3 (z)(dz/z) = e 4πkδ r 3 (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. 5]) hence k=0 θ(e 2πih/k 2πz ) = (k 2z) G(h, k) +..., θ(e 2πih/k 2πz ) 6 = (8k 6 z 3 ) G(h, k) , where G(h, k) = k j=0 e2πihj2 /k is the Gaussian sum. Now, (2π) θ(e 2πih/k 2π(δ i(ξ/2π)) ) 6 dξ = (8k 6 ) G(h, k) 6 δ iξ 3 dξ +... = (8k 6 ) G(h, k) 6 δ 2 = δ 2 (4k 6 ) G(h, k) , (x 2 + ) 3/2 dx +... hence S = (2δ) 2 h,k k 6 G(h, k) , so that (2.) S = (2δ) 2 A k +... k=

9 with A k = k 6 Lattice points in a sphere 39 h mod k;(h,k)= G(h, k) 6. The dots in (2.) stand for o(δ 2 ), δ 0. A k is a multiplicative arithmetical function (see [Gro, p. 56]), so that A k k 2 = A k A k2 when (k, k 2 ) =. Hence (2.2) ( + A p + A p ) k= A k = p with the product over primes. Now, A 2 = 0 and for a >, G(h, 2 a ) = 2 (a+)/2 (see [Gro, p. 38]), hence A 2 a = 2 6a 2 3(a+) 2 a = 2 2a+2, a >, and + A 2 + A 4 + A = + /3 = 4/3. When p > 2, G(h, p a ) = (h/p a ) G(, p a ) = p a/2 (see [Gro, p. 38]), hence A p a = p 6a p 3a h = (p )p 2a and + A p + A p = + (p(p + )). Therefore from (2.) and (2.2), S = (2δ) 2 4 { + (p(p + )) } +... = δ 2 2 { + (p(p + )) } p>2 Now, p { + (p(p + )) } = ζ(2)/ζ(3), hence (2.3) S = δ Thus and so lim δ 0 δ2 lim δ 0 δ2 ζ(2) ζ(3) +... r 3 (k) 2 exp( 4πkδ) = 2 7 k= k= p ζ(2) ζ(3) r 3 (k) 2 exp( kδ) = 32 ζ(2) π2 7 ζ(3). By the tauberian theorem of Hardy and Littlewood [HL] this implies that N lim N 2 r 3 (k) 2 = 6 ζ(2) π2 N 7 ζ(3). Hence by partial summation, k= lim (log N) N N r 3 (k) 2 k 2 = 32 7 k= π2 ζ(2) ζ(3).

10 392 P. M. Bleher and F. J. Dyson Acknowledgements. The authors thank Fernando Chamizo and Henryk 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. Adhikari and Y.-F. S. Pétermann, Lattice points in ellipsoids, Acta Arith. 59 (99), [Ble] P. M. Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math. J. 67 (992), [Ble2], Distribution of energy levels of a quantum free particle on a surface of revolution, ibid. 74 (994), [BCDL] P. M. Bleher, Z. Cheng, F. J. Dyson and J. L. Lebowitz, Distribution of the error term for the number of lattice points inside a shifted circle, Comm. Math. Phys. 54 (993), [BK] M. N. Bleicher and M. I. Knopp, Lattice points in a sphere, Acta Arith. 0 (965), [CI] F. Chamizo and H. Iwaniec, A 3-dimensional lattice point problem, in preparation. [CN] K. Chandrasekharan and R. Narasimhan, Hecke s functional equation and the average order of arithmetical functions, Acta Arith. 6 (96), [Che] J.-R. Chen, Improvement on the asymptotic formulas for the number of lattice points in a region of the three dimensions (II ), Sci. Sinica 2 (963), [Cra] H. Cramér, Über zwei Sätze von Herrn G. H. Hardy, Math. Z. 5 (922), [Gro] E. Grosswald, Representations of Integers as Sums of Squares, Springer, New York, 985. [HL] G. H. Hardy and J. E. Littlewood, Tauberian theorems concerning power series and Dirichlet series whose coefficients are positive, Proc. London Math. Soc. 3 (94), 74. [H-B] D. R. Heath-Brown, The distribution and moments of the error term in the Dirichlet divisor problem, Acta Arith. 60 (992), [KN] E. Krätzel and W. G. Nowak, Lattice points in large convex bodies, II, ibid. 62 (992), [Lan] E. Landau, Vorlesungen über Zahlentheorie, V., Hirzel, Leipzig, 927. [Lan2], Ausgewählte Abhandlungen zur Gitterpunktlehre, A. Walfisz (ed.), Deutscher Verlag der Wiss., Berlin, 962. [Now] W. G. Nowak, On the lattice rest of a convex body in R s, II, Arch. Math. (Basel) 47 (986), [Ran] B. Randol, A lattice point problem, I, II, Trans. Amer. Math. Soc. 2 (966), ; 25 (966), 0 3. [Sze] G. Szegö, Beiträge zur Theorie der Laguerreschen Polynome. II : Zahlentheoretische Anwendungen, Math. Z. 25 (926), [Vau] R. C. Vaughan, The Hardy Littlewood Method, Cambridge University Press, Cambridge, 98. [Vin] I. M. Vinogradov, On the number of integral points in a given domain, Izv. Akad. Nauk SSSR Ser. Mat. 24 (960),

11 Lattice points in a sphere 393 [Vin2] I. M. Vinogradov, On the number of integral points in a sphere, ibid. 27 (963), [Wal] A. W a l f i s z, Gitterpunkte in mehrdimensionalen Kugeln, PWN, Warszawa, 957. DEPARTMENT OF MATHEMATICAL SCIENCES SCHOOL OF NATURAL SCIENCES INDIANA UNIVERSITY INSTITUTE FOR ADVANCED STUDY PURDUE UNIVERSITY INDIANAPOLIS PRINCETON, NEW JERSEY BLACKFORD STREET U.S.A. INDIANAPOLIS, INDIANA U.S.A. BLEHER@MATH.IUPUI.EDU Received on and in revised form on (2584)