A multidimensional analog of the Weierstrass -function in the problem of the number of integer points in a domain Текст научной статьи по специальности «Математика»

УДК 517.55

A Multidimensional Analog of the Weierstrass (-function in the Problem of the Number of Integer Points in a Domain

Elena N. Tereshonok* Alexey V. Shchuplev^

Institute of Mathematics, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 21.03.2012, received in revised form 21.04.2012, accepted 15.05.2012 A multidimensional analog of the Weierstrass Z-function in Cn is a differential (0, n — l)-form with singularities in the points of the integer lattice Г С Cn. Using this form we construct a Г-invariant (n,n — l)-form т(z) Л dz. The integral of this form over a domain's boundary is equal to difference between the number of integer points in the domain and its volume.

Keywords: Weierstrass Z-function, integer lattice, Bochner-Martinelli kernel, Gauss circle problem.

The number of integer lattice points in a circle centered at the origin equals its area plus some error term. There is a problem of estimating the growth of the error as the radius of the circle increases. This question was posed and answered by C.F. Gauss who proved that the difference of the number I of integer points in the circle x2 + y2 ^ r2 and its area V is equal to O(r). Some of the great mathematicians of the last century spent time trying to improve this estimate: Sierpinski, Hardy, Littlewood, Vinogradov to name but a few. The best known result is due to M. Huxley stating that the error is O(r 73), although it is conjectured that it must be O(r 2 +e) (see [1] or [2] for the discussion). The same question can be obviously asked about any domain and not only planar ones (see e.g. [3-5]).

If D is a domain in C = R2 then to count its integer points one can use the Weierstrass Z-function for the lattice Г of Gaussian integers

z (z) = (+ ± + 4) •

z z" Y Y Y2/

If dD is piecewise smooth and does not intersect Г then the number of the lattice points in D is

I (D)=2bJ Z dz.

dD

It was shown in [6] that while Z(z) is not Г-invariant this may be corrected by adding a linear term, and its integral over dD gives then the desired difference

I (D) - Vol (D) = ¿r J (Z (z) - nz) dz. (1)

dD

* l.tereshonok@gmail.com t alexey.shchuplev@gmail.com © Siberian Federal University. All rights reserved

To follow this idea in higher complex dimensions, one should use then the representation of the Dirac ¿-function by the Bochner-Martinelli integral and an analog of the Weierstrass Z-function [7].

Let r = (Z + iZ)n be an integer lattice in Cn, denote by wBM the (0, n — 1) differential form

n fc-1

( — 1) zk d [Zk]

, \ _ k=1 wBM (z) = -—pn-•

iNI

Following [7], we define an analog of the Weierstrass Z-function as the differential form

n 'd^BM , ^ dw

Z (z) = wbm (z) + wbm (z — Y) + wbm (y) + (

7er' V i=i ^

BM , , dWBM , _

(Y) zi + a- (Y) Z

^ \ • 1 x dzi dz

76r' \ i=1 v

where r' = r \ {0} • This series converges absolutely and uniformly on compact subsets of Cn \ r. As the classical Z-function, this form is not r-invariant, but we can correct this.

Lemma. There exists a differential (0, n — 1)-form l (z) with linear coefficients such that t (z) = Z (z) — l (z) is r-invariant.

Proof. Let w = (wi, w2, • • •, wn) = (xi + iyi, x2 + iy2, • ••,xn + ¿yn) be an element of the lattice r, for z / r we consider the difference

Z (z + w) — Z (z) • (2)

While it is not r-invariant, the derivatives of its coefficients with respect to zi and zi are, as is shown by a simple computation. Therefore, this differential form depends only on w and w. To compute (2) we treat each its coefficient separately. Write

n

Z (z) = £ Zkdz[k], k =

k=1

then

z+w

Zk (z + w) — Zk (z) = J dZk, k = 1, n

z

These integrals do not depend on the path connecting z and z + w as long as it does not pass through the points of the lattice r. So we take z such that none of z1, z2, • • •, zn is a Gaussian integer, and a polygonal chain consisting of line segments connecting points (z1, z2, • • •, zn), (z1 + X1, z2, •••, zn), (z1 + w1, z2, •••, zn) , (z1 + wj, z2 + X2, z3, •••, zn),

(z1 + w1, z2 + w2, z3, •••, zn)v • •, (z1 + w1, •••, zn + xn), (z1 + w1, •••, zn + wn) as the integration path.

The integral of dZ1 along the first segment of the path equals

xi

J (d^ (z1 + t, z2, •••, zn) dt + Iz"! (z1 + t, z2, •••, zndt,

0

which is equal to

1

X1 J (z1 +1, z2, •••,zn) + d=1 (z1 + t,z2,:,zndt,

due to r-invariance of the derivatives of Zi- The similar holds for the rest. Thus, we can write

n n

Z (z + w) - Z (z) = Xkamkdz [m] + iykßmkdz [m].

k=1 m=1

where

amk = J (^oZm (zi, •••, zk +1, ..., Zn) + dm (zi, •••, zk +1, ..., zn)^ dt, 0 i

pmk = J ^dzk (zi' •••' zk + it, •••, zn) - (zi, •••, zk + it, ..., z„)^ dt.

0

Computing the derivatives of Zk , k = 1, n, we see that the coefficients amk and fimk are related as follows

^mk = -iamk, k = m, ^mk = -amk, k = m.

Therefore we have

Z (z + w) - Z (z) = ^ ^ [Jmfcwfcamfc + (1 - ¿mfc) (xk + yk) amfc] dz [m] ,

k = 1 m= 1

where £mk is the Kronecker delta. It is clear that we may take

n n

l (z) = "^2 "^2 №mkZkamk + (1 - 5mk) (Rezk + Imzk) amk] dz [m] .

k=1 m=1

□

The form Z(z) — l(z) is T-invariant, we shall denote it by t(z). Similarly to the one-dimensional (planar) case, there is a version of formula (1).

Theorem. Let D be a domain in Cn with a piecewise smooth boundary dD without lattice points on it. Then the number of lattice points inside D is related to the volume Vol (D) by the formula

I - Voi(D) = —Ln /" t (z) A dz. (2ni) J

dD

Proof. The integration set is obviously homologous to the sum of spheres of sufficiently small radius with centers in the lattice points inside D. Let S be one of such spheres. The integral

l0 \n I Z (z) a dz (2ni) J

S

over the sphere with the center w = (w1, ..., wn) e r equals to 1. Indeed, the interior of S = dB contains only one term of Z (z) a dz with singularity: wBM (z — w) a dz. Due to Stokes' theorem, the remaining terms are either zero or cancel each other. Therefore,

1 r Z (z) a dz (3)

(2ni)n J

dD

nn

is equal to the number of the lattice points in the interior of D. Consider now

fo \n I l (z) a dz. (2ni) J

dD

The integrand does not have any singularity in D, therefore again by the Stokes' theorem we have

-n l (z) adz = — y] y]d [Smkwkamk + (1 - 5mk) (Rewfc + Imwfc) amk] dz [m] =

0 J (2ni) J k= m=

d D D k=1 m=1

(2ni)n J w (2ni)

D

1 f "

V (-1)m-1 ammdz A dz.

D m=1

Eventually,

1 fl (z) A dz = VnD)E(-1)m-1 an

(2ni)n J n

dD m=1

To determine the value of m= ( —1)m-1 amm let us consider D to be a 2n-dimensional hyper-cube Q with the center at the origin with edges of length 1 parallel to coordinate axes. When integrating t (z) along boundary of D, the integrals over opposite faces of the hypercube cancel each other due to the r-invariance of t(z) . Thus,

-1-— i t (z) a dz = 0. ni) J

(2ni)

dD

On the other hand, there is only one integer point, the origin, inside of Q. Therefore,

1 Vol(D) ^ ( 1)m-1 0

1--—2-^ (-1) amm = 0,

m= 1

and

n

\m-1

(-1)

am

This proves the theorem. □

Remark 1. Denote the basis vectors of the fundamental parallelepiped of r by Yj, j = 1,2n: for j < n Yj = (y 1j,..., Ynj) such that ykj = ^kj, k = 1, n and for n < j < 2n Yj = iYj-n- Then in the notation of [7, p. 92] we have

) akj, j < n,

nkj = i

[ipkj, n < j < 2n.

In the case of r = (Z + iZ)2 c C2 we have a11 — a22 = n2. The computer experiments show that

n2

an = —^11 = —a22 = P22 = , a12 = ^12 = a21 = #21 = 0.

Remark 2. Notice that formula (3) gives the number of integer lattice points in a domain D in the even dimensional space R2n = Cn. However, it can be used for domains in the space of odd

1

dimension R2n+1 too. Let G be a domain in R2n+1, in order to apply (3), we complexify R2n+1 and define D to be the domain D = {G + -1}, where Ie = [—e, e] с R.

This paper was supported by Ministry of Education and Science of Russian Federation grant

(№1.34.11). References

[1] R.K.Guy, Unsolved problems in number theory, Springer-Verlag New York, 1994.

[2] S.E.Cappell, J.L.Shaneson, Some problems in number theory I: the circle problem, http://arxiv.org/pdf/math/0702613v3.pdf, 2007.

[3] L.A.Aizenberg, Application of multidimensional logarithmic residue to represent the difference between the number of integer points in a domain and its volume in the form of an integral, Dokl. Nauk SSSR, 270(1983), no. 3, 521-523 (Russian).

[4] E.Kratzel, Lattice points, Mathematics and its Applications (East European Series), 33(1988).

[5] A.N.Varchenko, Number of lattice points in families of homothetic domains in Rn, Funkts. Anal. Prilozh., 17(1983), no. 2, 1-6 (Russian).

[6] R.Diaz, S.Robins, Pick's Formula via the Weierstrass p-function, The American Mathematical Monthly, 102(1995), no. 5, 431-437.

[7] P.Zappa, Sulle classi di Dolbeault di tipo (0,n — 1) con singolarita in un insieme discreto, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 8(1981), no. 70, 87-95 (Italian).

Многомерный аналог Z-функции Вейерштрасса в задаче о числе целых точек в области

Елена Н. Терешонок Алексей В. !Щуплев

Многомерный аналог Z-функции Вейерштрасса в Cn есть дифференциальная (0, n — 1)-форма с особенностями в узлах целочисленной решётки Г С Cn. В статье при помощи этой формы строится Г-инвариантная (n, n — 1)-форма т(z) Л dz, интеграл которой по границе области D С Cn равен разности числа целых точек в области и её объёма.

Ключевые слова: Z-функция Вейерштрасса, целочисленная решётка, ядро Бохнера - Мартинелли.