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УДК 517.55
An Integral Formula for the Number of Lattice Points in a Domain
Lev Aizenberg*
Department of Mathematics Bar-Ilan University 52900 Ramat-Gan Israel
Nikolai Tarkhanov^
Institute of Mathematics University of Potsdam Am Neuen Palais, 10, Potsdam, 14469
Germany
Received 06.02.2015, received in revised form 06.03.2015, accepted 14.04.2015 Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of Rn and the volume of this domain. The difference proves to be the integral of an explicit differential form over the boundary of the domain.
Keywords: logarithmic residue, lattice point.
Introduction
Classical function theory is of great importance in number theory, let alone the analytical extension of the Riemann zeta function and prime number theorem, see [6,8,9], etc.
This work was intended as an attempt at applying the theory of functions of several complex variables to classical problems of number theory. To wit, we apply the multidimensional logarithmic residue which is an efficient numerical tool of algebraic geometry, see [1].
Let Z be a bounded domain with piecewise smooth boundary in the space Cn of n complex variables z = (z1,... ,zn). Consider a holomorphic mapping w = f (z) of the closed domain Z into Cn which has no zeros at the boundary of Z. Then f has only isolated zeros in Z and the number of zeros counted with their multiplicity is given by the logarithmic residue formula
N(f, Z) = f ^jf^r ¿(-^flf] A df (1)
Jez (2ni)n j= |fI
(see [1, § 2]), where |f |2 = |fi|2 + .. . + |fn|2, fj being the j th component of f, by df = dfiA.. .Adfn is meant the exterior product of the differentials df1,... ,dfn, and df [j ] stands for the exterior product of the differentials df,..., dfn after each other, the differential df j being omitted. The domain Z is oriented in such a way that
f 1
-——dz A dz > 0.
Jz (2i)n
We apply formula (1) to get an equality for the difference between the number of lattice points in the domain Z and its volume. A number of classical problems of number theory, e.g. the
* aizenbrg@gmail.com ttarkhanov@math.uni-potsdam.de © Siberian Federal University. All rights reserved
problem on the number of lattice points in a ball [10], the problems on Dirichlet divisors [4], etc. reduce to evaluating asymptotics of the difference. It is worth pointing out that this asymptotics can not be found by standard methods, such as the Laplace method, stationary phase method, or saddle point method.
The theory of lattice points in large regions has attracted the interest of many mathematicians for more than eleven decades. The monograph [5] presents a broad survey of the main problems and results in lattice point theory.
1. The integral formula
As usual, we write Rn, n > 1, for the n-dimensional real Euclidean space of variables x = (xi,..., xn) with Xj e R. Suppose X is a bounded domain in Rn whose boundary is piecewise smooth and does not contain any point with integer coordinates. Denote by N(X) the number of integer points in X and by V(X) its volume.
Theorem 1. If the boundary dX does not contain lattice points then the difference N(X) — V(X) can be written in the form
___
^t[j ]sin(2nxj )vj
N (X) — V (X) = I ...I dtl ^~2(n — 1)1 —¡j-ds, (2)
/0 J0 hx " (^(t2 — 2tj cos(2nxj))+ n)"
j=i
where dt = dti A ... A dtn, t[j] = ti .. .tj-itj+i .. .tn, ds is the surface measure of dX and v(x) = (vi(x),..., vn(x)) is the unit outward normal vector of the boundary at x e dX.
Proof. Consider the domain Z = X x Y in Cn, where Y is a bounded domain with piecewise smooth boundary in the space Rn of variables y = (yi,... ,yn). We assume that 0 e Y. The points z = (zi,..., zn) of Z have the form Zj = xj + iyj, for j = 1,... ,n. As holomorphic mapping f : Z ^ Cn vanishing solely at the entire points of X x {0}, we take
fi (z) = e2nizi - 1, fn (z) = e2nlZn - 1,
each zero being simple. By formula (1), we get
J2(-1)j-i(e-2nlZ> - 1)de-2nlz[j] A de2nlz N f Z )=! (n -j=L- ,
JdZ (2m)n Me2n»zi _ 1|2 + ... + |e2™z„ _ 1|2\
where
de-2niz[j] = de-2nlZi A ... A de-2™^'-1 A de-2ntzi+i A ... A de-2nizn, de2nlz = de2nlZl A ... A de2nlzn. The right-hand side is easily reduced to
. / -2ni £ Zk -2niY, ZkN 2ni Y, Zk
2_^( — iy [e k=1 — e k=j Je k=1 dz[j] A dz f (—2ni)n-i(n— 1)! --^-. (3)
JdZ l |e2nizi — 1|2 + ... + |e2nlZn — 1|2 j
n
n
A trivial verification shows that
dz [j] A dz = (2i)n—1 ((-1)n-1dx A dy[j] + idx[j] A dy) (4)
for all j = 1,..., n. Using (4) one separates the real and imaginary parts of (3), these are
. / e yk E yk-2nyj
)n+J(e k=i _ e
. . / -4n ^ yk -4n z^ yk-2nyj \
2^(-1)n+^e k=i - e k= cos(2nxj )J dx A dy[j] (4nf-1 (n-1)! -^------+
)ez I ie2nizi _ 112 + + ie2nizn
n
(je2™zi - 1|2 + ... + |e2nlzn - 1|2J " ,-1 -4n E yk-2nyj
> (-1)j 1e k=j sin(2nxj)dx[j] A dy
(5)
+ / (4n)n-1(n-1)!
ZdZ
|e2nizi - i|2 + ... + |e2nizn - i|:
and
._ 1 / -4n E yk -4n E yk-2nyj \
2^(-1)j M e k=1 - e k=j cos(2nXj )j dx[j] A dy
f (4n)n-1(n-1)! j=-7--+
' dZ ( |e2nlzl - 1|2 + ... + |e2nlzn - 1|2 j
■A -4n E yk-2nyj
^(-1)n+j-1e k=j sin(2nxj)dx A dy[j]
\n — 1
+ f (4n)n—1(n-1)!
JdZ (
respectively.
The number N(f, Z) is real, hence it suffices to consider the mere real part (5) of formula (1). Moreover, we make the change of variables
t1 = e—2nyi,
t = e —2ny„
°n e ,
obtaining
^(-1)j 111 .. .tn (tj - cos(2nxj))dt[j] A dx
n
n
- 2tj cos(2nxj)) + nj
j=1
^(-1)n+j—1t[j] sin(2nxj )dt A dx [j]
n
n
(I>2 - 2tj cos(2nxj)) + nj
N(f, Z)= /' 2n—1(n-1)! -n-n-+
^^'t2 - 2tj cos(2nx,)) + nl
'j=1
+ / ^i-D! i=L^-= 11 + i2,
JdZ' n / — ~ Nn
(6)
j=1
where Z' is the image of the domain Z under the change of variables tj = e—2nyj, for j = 1,..., n.
This change involves the mere variables y whence Z' = XxT, where T is the image of Y by tj = e—2nyj with j = 1,... ,n. Since Y contains the origin, the n-tuple with coordinates 1 belongs to T. We now give the domain T the following concrete form
T = {t G Rn : r2 < |t|2 < R2} n {t G Rn : t1,..., tn > e},
n
n
where r < y/n, R > y/n and e > 0 is small enough. The boundary dT consists of a piece Sr of the (n — 1) -dimensional sphere {t G Rn : |t| = r}, a piece SR of the (n — 1) -dimensional sphere {t G Rn : |t| = R}, and pieces Hj of hypersurfaces tj = e parallel to the coordinates hyperplanes tj =0. According to this structure of the boundary of T we represent the integral I1 as the sum of integrals I1sSr, Ii,sR and I1jHj with j = 1,... ,n.
Let the piece H1 tend to the hyperplane {t1 =0}. At this hyperplane we obviously get
n n
Y(t* — 2tj cos(2nxj)) + n = Y(tj — cos(2nxj))2 + sin2(2nxj) + 1 > 1.
j=1 j=2
Therefore, the integral I1,h1 tends to zero as H1 tends to the hyperplane {t1 = 0}. Analogously, I1,Hj tends to zero as Hj tends to the hyperplane {tj = 0}, for each j = 2,...,n.
It remains to consider the limits of the integrals I1,Sr and I1,sr , when r ^ 0 and R ^ <x>. Let S"-1 be the part of the unit sphere with centre at the origin which lies in the cube 0 < tj < 1, j = 1,...,n. We endow S"-1 with the usual orientation, then Sr = —rS"-1 and SR = RS"-1. (When we tended Hj to the hyperplane {tj = 0} for all j = 1,... ,n, then Sr and SR became one 2n-th spheres.) Hence it follows readily that
n,sr
J2( — 1)j-1 r2n-1t1 ...tn (rtj — cos(2nxj ))dt[j]
i dx i 2n-1(n—1)\ 3—-n--> 0
X (r2 — 2 Y rtj cos(2nxj) + n)n
as r ^ 0.
On the other hand, we get
J2( — 1)j-1 R2n-1t1 ...tn (Rtj — cos(2nxj ))dtjj ]
n,sr = [ dx [ 1 2n-1(n — 1)!^--
>0 (R2 — 2^2 Rtj cos(2nxj)+ n)"
j=1
f dx f 2n-1(n — 1)! Y( — 1)j-111 ...tn tj dt[j],
Jx Jsn-1 r";
j=1
as R ^ <x. The last integral just amounts to V(X), for
f 2n-1(n—1)! Y( — 1)j-111 ...tn tj dt[j] = f 2n-1(n— 1)! t1 ...tn ds = 1.
j=1 Js^o1
Thus, if the domain T expands to the nonnegative one 2n -th space as above, the integral I1 tends to V(X). And the integral I2 converges to the integral on the right-hand side of formula (2), for dZ' = (dX xT) U (X x dT and ( — 1)j-1dx[j] = Vjds for all j = 1,..., n, as desired. □
For the most practical cases n = 2 and n = 3 Theorem 1 was first proved in [2].
2. The one-dimensional case
In this section we clarify the structure of formula (2) by directly computing the integral on the right-hand side of this formula in the case n = 1. Let X = (a, b), where m < a < m +1 and
n
—-
—5-
M < b < M + 1, m and M being integer numbers satisfying m < M. Then
I = i™ dt i 1 sin 2nx =
Jo Jsx 2n (t - cos 2nx)2 + (sin 2nx)2 = r1 / sin 2nb sin 2na \
= J0 2n V (t - cos 2nb)2 + (sin 2nb)2 (t - cos 2na)2 + (sin 2na)2 / .
Substituting s = t - cos 2nb and s = t - cos 2na into the first and second terms on the right-hand side, respectively, we get
f1 sin 2nb , f1 sin 2na I = --T-7-- --r----TTds =
./— cos 2-Kb 2n s2 + (sin 2nb)2 ,/— cos 2na 2n s2 + (sm2na)2
1 s 1 s = TT" arctan~ —— [_cos2nb arctan — - t-cos2na .
2n sin 2nb 2n sin 2na
To be specific, we consider the case
m +1/2 < a < m + 1,
M < b < M + 1/2,
then sin2na < 0 and sin2nb > 0. Hence it follows that
1 (n f cos2nb\ ( f cos2na
I = —--arctan--—--+ arctan--
2nV 2 V sin 2nb/ V 2) V sin2na
= — (n + arctan cot 2nb - arctan cot 2na). Finally, on using the equality arctan x = n/2 - arccot x we deduce
I = ^(n - arctan cot(2nb - 2nM) + arctan cot(2na - 2n(m + 1/2)) = (M - m) - (b - a), 2n
which just amounts to N(X) - V(X), as desired.
3. Some comments
It is easy to see that the integrations over t e [0, <)n and x e dX in formula (2) can be exchanged. In this way we get
n
N (X) - V (X )=/ V(-1)j—1Fj (x) sin 2nxj- dx[j ], (7)
Jax j=1
where
Fj(x) = f f -n-^-- *
0 0 ( X/(tfc - cos2nxfc)2 + ^(sin2nxfc)2j
k=1 k = 1
are functions of cos2nxj and sin2nxj, for j = 1,..., n. The differential form under the integral over dX on the right-hand side of (7) is smooth away from the lattice of half-integer points in Rn. As is seen from Section 2, the differential form is not closed outside this lattice. The coefficients Fj bear certain symmetry in variables x1,...,xn, perhaps, it suffices to compute only one of these coefficients in order to determine the others. Moreover, Fj can be computed in a closed
form, however, the expressions are cumbersome, cf. formula (3) in [3]. It is possible that formula (7) can be applied to construct asymptotics of the difference N(X) — V(X) as R ^ <, where X is the ball of radius R with centre at 0 or, more generally, an ellipsoid
or another expanding domain, cf. [5,7]. But we will not develop this point here.
This research was supported by the German Research Society (DFG), grant TA 289/4-2. The first author wishes to express his gratitude to Professor Sinai Robins who payed attention to the paper [2], invited the author to attend Singapore for two weeks and thus recommenced his interest to the topic.
References
[1] L.A.Aizenberg, A.P.Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, AMS, RI, 1983.
[2] L.Aizenberg, Application of multidimensional logarithmic residue for representation in the form of the integral of the difference between the number of integer points in a domain and its volume, Dokl. Akad. Nauk SSSR, 270(1983), no. 3, 521-523 (in Russian).
[3] L.Aizenberg, Application of multidimensional logarithmic residue in number theory, An integral formula for the difference between the number of lattice points in a domain and its volume, Annales Polonici Mathematici, 46(1985), 395-401.
[4] K.S.Chandrasekharan, Introduction to Analytic Number Theory, Springer, New York, 1968.
[5] F.Fricker, Introduction to Lattice Point Theory, Textbooks and Monographs in the Exact Sciences, Mathematical Series, 73, Birkhauser Verlag, Basel-Boston, Mass., 1982.
[6] Th.W.Gamelin, Complex Analysis, Springer, New York et al., 2001.
[7] R.K.Guy, Unsolved Problems in Number Theory, Springer, New York et al., 1981.
[8] D.J.Newman, Analytic Number Theory, Springer, New York et al., 1998.
[9] E.C.Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Lon-
[10] I.M.Vinogradov, Particular Versions of Trigonometric Sum Method, Nauka, Moscow, 1976 (in Russian).
Интегральная формула для числа целых точек в области
Используя формулу многомерного логарифмического вычета, мы даем простую формулу для разности между числом целых точек в ограниченной области из К" и объемом этой области. Эта разность дается интегралом от дифференциальной формы, задаваемой точным выражением, по границе этой области.
Ключевые слова: логарифмический вычет, целая точка.
don, 1951.
Лев Айзенберг Николай Тарханов
