Ekkehard Krätzel. (t 1,t 2 )+f 2 t 2. Comment.Math.Univ.Carolinae 43,4 (2002)

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1 Comment.Math.Univ.Carolinae 4, Lattice points in some special three-dimensional convex bodies with points of Gaussian curvature zero at the boundary Eehard Krätzel Abstract. We investigate the number of lattice points in special three-dimensional convex bodies. They are called convex bodies of pseudo revolution, because we have in one special case a body of revolution and in another case even a super sphere. These bodies have lines at the boundary, where all points have Gaussian curvature zero. We consider the influence of these points to the lattice rest in the asymptotic representation of the number of lattice points. Keywords: convex bodies, lattice points, points with Gaussian curvature zero Classification: P, H6. Introduction and statement of result Let F denote the distance function of the convex body PR. That is F t,t,t = { t + t + t }, PR = {t,t,t R : F t,t,t }. It is assumed that, N,, >, a divisor of. Then we have abodyofrevolution for = and a super sphere for =. Therefore, we call PR a body of pseudo revolution in general. We consider the points t,t,t at the boundary and we are confined to the points t,t,t without loss of generality. We put F t,t,t =, t = ft,t, where f is given by ft,t = t + t. The Gaussian curvature in such apoint is defined by K = Hft,t + f t t,t +f t t,t

2 756 E. Krätzel where Hft,t denotes the Hessian Hft,t = f t t t,t f t t t,t f t t t,t. In the present case we find by means of long but simple calculations K = t t t + t t. t +t + t From this it is seen that we have the following points with Gaussian curvature K = at the boundary: Body of revolution = : The curve t + t =,t = and the isolated points t,t,t =,, ±. Super sphere = : The curves t + t =,t =; t + t =,t =; t + t =,t =. Body of pseudo revolution <<: The curves t + t =,t =; t + t =,t =; t + t =,t =. The flat points t,t,t =,, ± are of exceptional importance in all three cases and the points t,t,t =±,,, t,t,t =, ±, are meaningful as well in the cases and. The aim of the paper is to investigate the number of lattice points in the dilated body of pseudo revolution x PR, that is: { } A, x;pr =# n,n,n Z : n + n + n x. Especially we study the influence of the points with Gaussian curvature zero to the asymptotic representation of A, x;pr. In [] a detailed description is given for the case of super spheres. See also the paper [5]. Therefore, we are here in the first place interested for the case <, but we do not exclude the case =. It is not too hard to obtain the following asymptotic representation for A x;pr from the results of the paper [7]: A, x;pr =volpr x +H,, x+h,, x+o x 5 +O x log x.

3 Lattice points in some special three-dimensional convex bodies The second main term H,, x is a certain function of x coming from the flat points t,t,t =,, ± and can be estimated by H,, x x. Analogously, the third main term H,, x is a certain function of x coming from the flat points t,t,t =±,,,, ±, and can be estimated by H,, x x. The first error term results from the other points with Gaussian curvature zero and the second error term results from the points with Gaussian curvature nonzero. In this paper we will give explicit representations of the second and third main terms which automatically show that the above upper bounds are at the same time lower bounds. Further we give an improved estimation of the first error term. Let the generalized Bessel functions J ν J ν x ν x = π ν + x be defined by t ν cos xt dt, where is the gamma function,, ν are real numbers with, ν>. Further let ν x = π ν + x ν J ν πnx, πn n= which is absolutely convergent for ν>. For a proof see []. Theorem. Let, N,, >, a divisor of. Then 4 A, x;pr =volpr x + H,, x+h,, x+, x with H,, x = / x =O, Ω H,, x =8x x, t t / xt dt = O, Ω x,, x x log x x log x.

4 758 E. Krätzel. Preparation of the problem We find, by symmetry, A, x;pr = 6S,, + S,, + S,, +Ox, where S i,j, are triple sums S i,j, = n n n with the summation conditions SCS i,j, : n i n j n, n + n + n x, n i =,n i = n j, n j = n get a factor. We begin the summation process in each sum with n i. For example, summing in S,, over n,we obtain [ x n ] n + for x n n <n and n otherwise. If we use [y]+ = y y, we get a term which can be written as an integral, and a term with the -function as a remainder. Hence Here 6S,, = S,, +,x+ox. S,, =6 n n t dt with the summation-integration conditions SIC : t n n, t + n + n x, and n n t 8, x = 6 n = n n,n D, gets a factor x n n,

5 Lattice points in some special three-dimensional convex bodies where D, denotes the domain { } D, = t,t R : x t t <t t. In the next step we sum over n in S,, and obtain similarly 6S,, = S,, + P,x+, x+ox with and 9 SIC n SIC t S,, =6 n t t t dt dt, : t t n, t + t + n x P, x = 6 n t n t : t n, x n t dt x n n <t x n. In the last step we sum over n in S,, and finally we obtain 6S,, = S,, + H,x+P, x+, x+ox, where S,, =6 dt dt dt, t t t IC : t t t, t + t + t x, t t t H, x = 6 t t x t + t IC = t t x t + t. t t dt dt,

6 76 E. Krätzel In the same way we obtain for the other triple sums: 6S,, = S,, + H,x+P, x+, x+ox, where S,, =6 dt dt dt, t t t IC : t t t, t + t + t x, t t t H, x = 6 t t IC : t t x t t t SIC x t t t, dt dt, P, x = 6 x n t t + n dt, n t, x = 6 D, = : t x t + n <n, n,n D, { t,t R : x n n, x t } t <t t. Finally, we obtain for S,, 8 6S,, = S,, + H,x+P, x+, x+ox, where 9 S,, =6 dt dt dt, t t t IC : t t t, t + t + t x, t t t

7 Lattice points in some special three-dimensional convex bodies H, x = 6 t t IC : t t t t SIC P, x = 6 n t n t : x t t dt dt, x t, x t n dt, x t <n x t t, x n + n,, x = 6 n,n D, { D, = t,t R : x t + t <t t }.. Representation of the triple integrals It is clear that it follows from, 4 and 9 which is the main term in 4. S,, + S,, + S,, =volpr x, 4. Representation and estimation of the double integrals We begin with H, x. We write the integration condition in in the form t t, t + t x t We have x t x.. We integrate only up to x. The remainder is of order x. Hence, by substituting t t x, t t x,we obtain H, x = 6x t t IC : t t, t + t t t. x t + t dt dt + Ox,

8 76 E. Krätzel Putting t + t = z we get, by symmetry, H, x = 8x z = 8 x = 8 x z x z z dz z x z dz + Ox z z z xz dz + Ox, z t dt + Ox where again the integral from up to z / / is of order x. By means of the Fourier representation of the -function, t = sinπnt dt, π n n= we obtain H, x = 8 x π = 8 x and, by and, n n= n= H, x = 4 π +x = / x+ox. z z sinπnxz dz + Ox z cosπnxz dz + Ox πnx J / πnx+ox n= Hence H, x =H,, x+ox and the representation 5 is obtained. The asymptotic representation of the generalized Bessel functions is given in Lemma. of []. We obtain { J / πnx = nx cos πnx π } πnx + + O. nx

9 Lattice points in some special three-dimensional convex bodies Hence, it follows with a positive constant c H,, x =cx n= Thus, the estimations in 5 are clear. n sin πnx π + Ox. In order to obtain the term H,, x we add H, X and H, x. Then we get from 5 and H, x+h, x = 6x IC : t t t x t t t t t dt dt,, t t t. By means of the substitution t = t / z we obtain H, x+h, x =6x IC : z t z t z xz dt dz, z t t z,t + z, t z. We may extend the domain of integration such that the new integration conditions are given by IC : t + z, t z t. The integral over the new domain is of order x which can be seen by partial integrating with respect to z. Now we substitute t = t. Then H, x+h, x = = 6x dt / The substitution z zt gives t t t z t z xz dz + Ox. H, x+h, x = = 6x t t dt z z xtz dz + Ox. /

10 764 E. Krätzel We tae the integral with respect to t from up to, since the new part of the integral is of order x. As in case of H,x we use the Fourier representation of the -function and we obtain analogously Hence H, x+h, x = = 6 π + x t + t =8x t t xt dt + Ox. / n J / n= H, x+h, x =H,, x+ox πnxt dt + Ox and the representation 6 is obtained. For the asymptotic representation we use 4. Clearly, the integral from up to is of order x. Therefore, we use the asymptotic representation of the generalized Bessel function from Lemma. in [] for t. Then J / πnxt = π πnxt Hence, it follows with a positive constant a H,, x = = ax n t + n= α cos πnxt π + + O. nx t sin πnxt π dt + Ox. The remaining integral has a singularity at t =. We obtain the asymptotic representation of the integral very easy by means of Chapter, Section from []. We use a special case of formula.6 on page 4: Let φt be continuously differentiable in α t β. Let φα =. Then, if <µ<, β e ixt β t µ φt dt = µ x µ eixβ µπi φβ+o. x The condition φα = is not necessary. In case of φα the point α yields an error term of order /x. Thus, with a constant b,we get H,, x =bx n= Hence, the estimations 6 follow immediately. n sin πnx π + Ox.

11 Lattice points in some special three-dimensional convex bodies Estimations of sums and integrals The aim of this chapter is to estimate the sums and integrals 9, 6 and. Lemma. Assume that for all z with x z and for all τ with τ x x n z,, x. x <n τ Then 5 P, x x,, x. Proof: It is easily seen that in 9 the condition t n is superfluous and that x n n > n > x. Further, consider 6 x n t dt, SIC : t x n n, x / + <n x. It is seen at once that this term is of order x. Now we bring 9 and this term together and obtain P, x = 6 x n t dt, Now we put t =x n / t. Then P, x = = 6 / SIC : t x n, x / + <n x. x / + <n x = 6I + I +Ox. x n x n t dt + Ox

12 766 E. Krätzel Applying partial summation we obtain by means of the condition of the lemma y I = x / + <n x xy,, x. x n x n t dt In I, where the integral is taen from y up to /, we use the Fourier expansion of the -function and obtain I = π x / + <n x t t sin = π m= m x n x / + <n x [ t t cos m= / t t m y πm x n t dt πm x n ] / t y / d t t cos πm x n t y dt } dt. We estimate the sum over n with van der Corput s simplest theorem see Theorem. in []. Then we get exactly in the same way as on page 8 of [] I x y. Now we put y = x,, x. Then the estimations of I and I are equal and we obtain 5. Lemma. Assume that for all z with z and for all u with x x n z,, x. x +z <n u x <u +z

13 Lattice points in some special three-dimensional convex bodies Then 6 P, x x,, x. Proof: The proof is quite the same as the proof to Lemma such that we omit it. Lemma. Assume that Then t <n t 7 P, x x Proof: Consider x 6 x t / t <n x t / t n,, x.,, x. x t n dt. It is easily seen that this term is of order x. Now we bring and this term together and obtain P, x = = 6 x Putting x t = t then P, x = 6 x t / <n x t / x t x t t <n t x t n dt + Ox. t n dt + Ox. Here we have the same situation as in Lemma 4.8 of []. Using this result 7 follows immediately. Estimations of the error terms,, x,,, x,,, x: G. Kuba [8] has pointed out that M.N. Huxley s [] method is applicable to the above sums. Assume that a, b, c, d are fixed positive real numbers. Then he proved X b / d<n X ac b / d X bn + d a 46 X 7 X ab log ab 5 46.

14 768 E. Krätzel But the proof shows essentially more: All the three cases in the lemmas are included. Hence P, x, P, x x log x 5 46, P, x x log x This gives the first term in the estimation Estimations of the double sums In order to estimate the sums 8, 7 and we apply Theorem in [4] or, what is the same, Satz 4.4 in [6]. For this purpose the domains D,, D, and D, must be divided into some subdomains. The technical realization is worst of all in case of D,. The both other cases are somewhat simpler and the calculations are in principle the same, but easier. Therefore, we consider only the estimation of, x, which is given by. Now let ft,t = x t + t such that f t t,t =t t + t x t + t. Let ν, ν be non-negative integers. Then we consider the following subdomains D, ν,ν ofd, : { D, ν,ν = t,t R : x t + t t t, It is easily seen that and from this x t + If a t b and c = b a then ν + f t ν + +, ν x t ν + x }. + ν t x t t ν x. c ν x. + ν +

15 Lattice points in some special three-dimensional convex bodies If α f t β and γ = α β then γ ν. We obtain for the second partial derivative and for the Hessian f t t = t t + t x t + t { t x + t x t + t } Hf =f t t f t t f t t 4 =t t t + t 4 x t + t {[ ] t x + t x t + t [ ] t x + t x t + t t t [ t + t + x t + t ] }. From this we get after some calculations f t t λ = ν + ν x, Hf Λ= ν + ν x. Now we apply Theorem from [4] or Satz 4.4 from [6]. The following assumptions of that theorem are certainly satisfied in the domain D = D, ν,ν : A Let D be a compact domain defined by D = {t,t :a σt t ϱt b,a t b }, with c = b a >, c = b a >, where c,c are so small as possible. Assume that σt, ϱt are partly monotonic and two times differentiable in [a,b ]. B Let ft,t be a real-valued function in D with continuous partial derivatives up to the third order. C Let f t t λ, Hf =f t t f t t f t t, Hf Λ.

16 77 E. Krätzel D Suppose that α f t β, γ = β α. E Let the function ϕy, t,t be defined by f t ϕy, t,t =y. Let ηt =ϱt, σt, ϕy, t. Then suppose that the functions η t, f t t ηt,tη t+f t t ηt,t, f t t ηt,tf t ηt,tη t are partly monotonic. Further, let ϕ y y, t be partly monotonic with respect to t. Then fn,n γ Λ 4 + λ c + λ n,n D { Λ + + c +γ + + λ } log Λ + log λ +. λ Λ Λ Now we use this result for our problem and we obtain, x = 6 n,n D, ν,ν x n + n ν ν + 4 x + ν + xlog ν + log ν + log x. We may sum over ν such that ν x, because the trivial estimation of the remainder gives an error term of order x /. Then, x x log x. Analogously we obtain the same estimation for, x and, x. This gives the second term in 7. References [] Copson E.T., Asymptotic Expansions, At the University Press, Cambridge, 965. [] Huxley M.N., Area, Lattice Points and Exponential Sums, Clarendon Press, Oxford, 996. [] Krätzel E., Lattice Points, Dt. Verlag d. Wiss., Berlin and Kluwer Academic Publishers, Dordrecht/Boston/London, 988. [4] Krätzel E., Double exponential sums, Analysis 6 996, 9. [5] Krätzel E., Lattice points in super spheres, Comment. Math. Univ. Carolinae , 7 9.

17 Lattice points in some special three-dimensional convex bodies [6] Krätzel E., Analytische Funtionen in der Zahlentheorie, B.G. Teubner, Stuttgart-Leipzig- Wiesbaden,. [7] Krätzel E., Lattice points in three-dimensional convex bodies with points of Gaussian curvature zero at the boundary, Monatsh. Math.,. [8] Kuba G., On sums of two -th powers of numbers in residue classes II., Abh. Math. Sem. Univ. Hamburg 6 99, Universität Wien, Institut für Mathemati, Strudlhofgasse 4, A-9 Wien, Austria Received January,, revised July,