PRIMITIVE LATTICE POINTS IN STARLIKE PLANAR SETS. Werner Georg Nowak. 1. Introduction. F (u) = inf

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1 pacific journal of mathematics Vol. 179, No. 1, 1997 PRIMIIVE LAICE POINS IN SARLIKE PLANAR SES Werner Georg Nowak his article is concerned with the number B D x of integer points with relative prime coordinates in x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for B D x. Applications to certain special geometric and arithmetic problems are discussed. 1. Introduction. Let D denote a subset of R 2 which is starlike with respect to the origin, i.e., if u R 2 belongs to D, automatically λu Dfor 0 <λ<1. he distance function F of D is defined by F u = inf { τ>0: } u τ D, with the usual understanding that inf =. Let us put Q = F 2, then Q is homogeneous of degree 2. For a large real variable x, we define A D x as the number of lattice points of Z 2 := Z 2 \{0, 0} in the blown up domain x D, i.e., ADx =# xd Z 2. We make the supposition that A D x satisfies an asymptotic formula 1.1 A D x = with R c r x αr + O x α, r=0 1.2 α 0 =1>α 1 > >α R >α, α<

2 164 WERNER GEORG NOWAK For a wealth of results of the form 1.1 on specific planar lattice point problems the reader may consult the monograph of Krätzel [11]. he objective of the present paper is to study the number B D x ofprimitive lattice points in xd, i.e., B D x =# { m=m 1,m 2 Z 2 : m xd, gcd m 1,m 2 =1 }. By Möbius inversion, 1.3 B D x = x µma D, m 2 m N where µm denotes the Möbius function. By an elementary convolution argument, one can derive from the bound 1.4 µm YωY m Y see Ivić [8, p. 309], combined with 1.1 and 1.3, 1.5 B D x = c r + O x 1/2 ωx, ζ2α r xαr where r: α r 1 2 ωx = exp clog x 3/5 log log x 1/5 with c>0, is a factor familiar from the Prime Number heorem. 1.4 and 1.5 contain the strongest information available to date concerning zero-free regions of the Riemann zeta-function. At the present state of art, it is not possible to reduce the exponent 1 of x in the O-term of 1.5. his will 2 be evident from Lemma 1 below with y = 1, in view of the fact that the Riemann zeta-function could have zeros with real part arbitrarily close to 1. It is therefore natural to search for stronger estimates assuming the truth of the Riemann Hypothesis henceforth quoted as RH. his problem has been attacked by Moroz [15], for the slightly simplified case that R = 0. He obtained the result that 6 B D x =c 0 π 2x+O x 2 α +ε 5 4α ε>0, conditionally under RH. We remark that recently Hensley [5] has recently written a paper on the subject, too, apparently unaware of Moroz s work. He used a methodically original approach but failed to sharpen the estimate. In this paper, our ultimate goal will be to prove the following.

3 PRIMIIVE LAICE POINS 165 heorem. Suppose that D R 2 is starlike with respect to the origin and that A D x satisfies the asymptotic formula 1.1. IfRH is true, B D x = r: α r>θ c r ζ2α r xαr + O x θ+ε, for a large real variable x, arbitrary fixed ε>0, and θ := 4 α 11 8α. Before going into technical details which we postpone to Sections 2 and 3, we outline the essential ideas of the proof. First of all, it will be convenient to consider the quantities and A Dx :=#{m Z 2 : Qm x}, B Dx :=#{m=m 1,m 2 Z 2 : Qm x, gcd m 1,m 2 =1}, instead of A D x, B D x. Since, for every δ>0, A D x A Dx A D x + δ, B Dx δ B D x B Dx, A Dx satisfies the asymptotic formula 1.1 as well, and the heorem is immediate for B D x if it has been proved for B Dx. Further, it is clear that Q 1 := inf Qu > 0. u Z 2 hus we may restrict the summation in 1.3 to1 m x/q 1, and obtain by splitting up 1.6 BDx = m yµma x D + x µma m 2 D =: S m m>y S 2, where y = yx < x/q 1 is a parameter remaining at our disposition. By 1.1, 1.7 S 1 = R c r x αr r=0 m y µm m 2αr + O x α y 1 2α.

4 166 WERNER GEORG NOWAK Using the classic conditional bound valid under RH 1.8 µm Y 1/2+ε ε > 0, m Y summation by parts gives µm 1 ζ2α = r y + O 1 2αr+ε 2 if α r > 1, 4 m 2αr O y 1 2αr+ε 2 else. m y hus 1.7 may be rewritten in the form µm S 1 = c r x αr + m 2αr r: α r> α m y r: α r α O r: 1 4 <αr α x αr y 1 2 2αr+ε + O x α y 1 2α. c r ζ2α r xαr o deal with S 2, an obvious possibility is to use 1.8 one more time and to apply summation by parts repeatedly. Observing that A D w is monotone and w, one obtains S 2 x ε y 1/2 x y 2. See Moroz [15, formula 8]. he key step of the present paper is to improve this elementary estimate by a contour integration technique in the spirit of a classic paper due to Montgomery and Vaughan [14]. Proposition. If the Riemann Hypothesis is true, S 2 = x µma µm D = c m m>y 2 r x αr m r: α r> α m>y 2αr + O x α+ε 2+α xy + O x ε y 1/ ε > 0, 2 for large real parameters x and y with 1 y x 1/2. We combine this result with 1.9 and note that the last O-terms in 1.9 and 1.10, respectively, are of the same order apart from ε s for 1.11 y = x 41 α 11 8α. his choice of y readily yields the assertion of our heorem, since it is easily verified that, for α r α, 3 x αr y 1 2 2αr x θ.

5 PRIMIIVE LAICE POINS Some Lemmas. For Re s>1, we define the zeta-function Z D s of the set D by the absolutely convergent Dirichlet series Z D s = Qm s. m Z 2 : Qm< We further put, for real y 1 and a complex variable s, 2.1 f y s = 1 ζs µm m. s m y his is regular in every s C which is not a zero of the Riemann zetafunction. Lemma 1. For a large real variable x, and any fixed C 5, S 2 = x µma D m m>y 2 = 1 3+ix C Z D s f y 2s xs 2πi 3 ix s ds + O x α+ε C ε>0, uniformly in 1 y x. Proof. his clearly is a type of truncated Perron s formula. It is hard to find an explicit reference in the literature, although the argument runs on familiar lines: Let us write the positive and finite values attained by Qm, as m runs through Z 2, in form of a strictly increasing sequence λ k k N. Put further { µm if m>y, µ y m = 0 else, then it follows by the homogeneity of Q that, for Re s>1, 2.2 Z D s f y 2s = n Z 2 : Qn< γnqn s = k=1 η k λ s k, with γn := m nµ y m, η k := n: Qn=λ k γn.

6 168 WERNER GEORG NOWAK Here m n 1,n 2 means that m gcd n 1,n 2. For later reference, we note that, for any n =n 1,n 2 Z 2 with Qn <, 2.3 gcd n 1,n 2 Qn. o realize this, let n := n 1 gcd n 1,n 2, n 2 gcd n 1,n 2, then n, 2n,...,gcd n 1,n 2 n all belong to 2QnD Z 2. herefore, by 1.1, gcd n 1,n 2 A D 2Qn Qn. Furthermore, 2.4 S 2 = µm m>y Qn x m 2 1 = m,n: Qmn x µ y m = It is well-known that, for a>0,a 1, and sufficiently large, 1 3+i 2πi 3 i a s s ds = χa+o O a 3, a 3 log a, k: λ k x where χ is the characteristic function of the interval ]1, [. Of this formula, may be found in Apostol [1, p. 243], while is immediate by taking as a path of integration the boundary of the domain {s C : s, Re s 3} if a>1, resp., of {s C : s, Re s 3} if a<1 cf. Prachar [21, p. 379]. herefore, by 2.2 and 2.4, ix C Z D s f y 2s xs 2πi 3 ix s ds C = S 2 + k: λ k x 1 O By the mean-value theorem, η k λ 3 kx 2 log λ k log x + k: λ k x <1 η k. O η k. log λ k log x 1 max λ k,x λ k x λ kx λ k x,

7 PRIMIIVE LAICE POINS 169 thus the first error term sum here is 1 x k=1 η k λ 2 k 1, since the series in 2.2 converges absolutely for Re s>1. Further, η k γn λ ε k n: Qn=λ k n: Qn=λ k 1, for any ε > 0, in view of 2.3 and the definition of γn. hus the second error term sum in 2.5 is x ε n: Qn x <1 in view of 1.1. his proves Lemma 1. 1 x ε A Dx+1 A Dx 1 x α+ε, he key point to prove the Proposition will be to have at hand the following estimates for the growth of the complex function Z D s in the vertical direction. Lemma 2. i For any σ 1 >α, there exists a positive real number ω<1such that Z D σ + it t ω, uniformly in σ σ 1, t 1. ii For a real parameter 4, fixed ε > 0, and any fixed β with 1 2+α β<1, it follows that 3 2 Z D β + it dt 1+ε. Proof. Let us rewrite 1.1 in the form 2.6 P Dx :=A Dx R c r x αr x α. Let further X denote a positive real number which is not attained by Qn as n runs through Z 2. Using Stieltjes integral calculus, we conclude that, r=0 1 For β 1, the estimate is trivial and not needed later.

8 170 WERNER GEORG NOWAK for Re s>1, 2.7 Z D s = = 0<Qn X Qn s = R c r α r w s+αr 1 dw + r=0 R r=0 c r X X α r s α r X αr s X s P D X+s X R w s d c r w αr + P D w r=0 w s dp D w X w s 1 P Dwdw. In this identity we choose 0 <X<Q 1 and let X Q 1 to obtain 2.8 Z D s = R r=0 c r s Q αr s 1 + s w s 1 P D w dw. s α r Q 1 In view of 2.6, this provides a meromorphic continuation of Z D s tothe half-plane Re s>α, with simple poles at s = α r,r=0,...,r. At the same time, 2.8 shows that Z D σ + it t, uniformly in σ σ 0, t 1, where σ 0 >αis arbitrary but fixed. Since, by absolute convergence, Z D σ + it is uniformly bounded in every half-plane σ σ 2 > 1, a Phragmén-Lindelöf argument 2 establishes part i of Lemma 2, ifweputσ 0 = 1α+σ 2 1 for arbitrary given σ 1 >α. o show ii, we apply the identity derived in 2.7 one more time, with 2.9 ξ X 2 ξ, ξ := 3 21 α, s = β + it, t 2. his is clearly justified by analytic continuation. We obtain 2.10 with Z D β + it =S X t+o 1 X 1 β +O X β+α, S X t := m Z 2 : Qm X Qm β it. 2 For a classic reference, see Landau [13, p. 229].

9 PRIMIIVE LAICE POINS 171 Integration over t 2 gives Z D β + it dt S X t dt + O X 1 β + O 2 X β+α. By Cauchy s inequality 3, 2 2 S X t dt 2 S X t 2 dt Qm Qn X Qm Qn β 2 it Qn dt Qm. For Qm <Qn, the integrals in this sum can be estimated by 2 exp itlog Qn log Qm dt 2 log Qn log Qm Along with the trivial bound, this gives 2Qn Qn Qm S X t dt m: Qm Qn Qn X Qn β 1 1 Qm max β, Qn Qm. Qn We now keep n Z 2 fixed for the moment and split up the inner sum over m: For that purpose, we define a sequence δ j J by δ j=0 j =2 j Qn 1, with J such that 1Qn <δ 8 J 1 Qn. We distinguish three cases according to 4 the relative size of Qn Qm. First of all Case 1, Qn Qm <δ 0 1 Qn Qm > Qn Qm >Qn 1 1, 3 Here and in what follows, m and n denote elements of Z 2.

10 172 WERNER GEORG NOWAK thus the contribution of these m to the inner sum in 2.12 is Qn β A D Qn A D Qn Qn 1 β + Qn β+α. Further Case 2, for 0 j J, Qn Qm [δ j, 2δ j [ Qn 2δ j <Qm Qn δ j, thus the corresponding portion of the inner sum in 2.12 is Qn1 β A D Qn δ j A D Qn 2δ j δ j Qn 1 β + δ 1 j Qn 1 β+α. Summing this over j = 0,...,J gives 2.14 O JQn 1 β + O δ 1 0 Qn 1 β+α ε Qn 1 β + Qn α β. Finally Case 3, the portion of the inner sum in 2.12 corresponding to the m s with Qn Qm 2δ J is 2.15 m: Qm Qn Qm β = Qn 1 2 u β da D u Qn 1 β. We now combine the upper bounds 2.13, 2.14, and 2.15, and use them in 2.12 to conclude that 2 2 S X t dt Qn X Combining this with 2.11, we obtain 2 Qn β ε Qn 1 β + Qn α β X = 1+ε w 1 2β da D w ε X 2 2β Z D β + it dt 1/2+ε X 1 β + + X 1 β + 2 X β+α. It is easy to see that the choice of X according to 2.9 is optimal, and that the bound obtained is 1+ε for β 2+α. hus the proof of Lemma 2 is 3 complete.

11 PRIMIIVE LAICE POINS 173 Lemma 3. If RH is true, the function f y s defined in 2.1 satisfies f y σ + it y 1 2 σ+ε t ε +1 ε >0 fixed, uniformly in σ 1 σ σ 2, y 1, for arbitrary σ 2 >σ 1 > 1 2. Proof. his key lemma of the Montgomery-Vaughan method is meanwhile well-known. See, e.g., Nowak and Schmeier [20], or Baker [2, Lemma 1]. We put 3. Proof of the Proposition. 3.1 β := 2+α +ε 3 with ε 0 as small as we please, such that 4 β/ {α 0,...,α R }. We start from Lemma 1 and shift the line of integration to Re s = β, applying the residue theorem. In view of clause i of Lemma 2 and Lemma 3, the horizontal segments contribute 3 x 3 C Z D σ + ix C f y 2σ +2ix C dσ x 3 C+Cω+ε 1 β for C sufficiently large. Furthermore, by clause ii of Lemma 2 and Lemma 3, β+ix C Z D s f y 2sx s ds β ix s C 2 x β y 1 2 2β+ε 1+ ε 1 Z D β + it dt =2 j x C, j=1,2,... x β+2cε y 1 2 2β+ε. Collecting results, we arrive at S 2 = Res s=αr Z D s f y 2s xs + O x α+ε + O r: α r>β s Since, by 2.8, Res s=αr Z D s f y 2s xs = c r x αr s m>y 2+α xy x ε y 1/ µm m 2αr for α r >β, this completes the proof of the Proposition and thereby that of our heorem. 4 ε is only needed to deal with the case that 2+α 3 is equal to one of α 0,...,α R.

12 174 WERNER GEORG NOWAK 4. Some applications to special problems Convex domains with nonzero curvature of the boundary. he most generic example is probably a convex planar domain D whose boundary D is sufficiently smooth 5 and has nonzero curvature throughout. We further suppose that the origin is an inner point of D. Under these conditions, a very deep and rather recent result of Huxley [6] says that 4.1 A D x = areadx + O x log x 146. Using this with our heorem, we obtain for the number of primitive lattice points in xd conditionally under RH, B D x = 6 π 2areaDx + O x 269 +ε 619. However, for this special problem, Huxley and the author [7] have established the better error term O x ε 269. Numerically, = , while = his result does not depend on 4.1, but was derived 12 using the mean-square bound x A D u areadu 2 du x 3/2. 0 For the case that D is the unit disk or any origin-centered rational ellipse, a recent idea of Baker [2] can be modified to prove under RH that B D x = 6 π x+o x 3 8 +ε. See also [7] for a bit more details Sums and differences of relative prime k-th powers. For a fixed natural number k 3, we ask for the average order of the arithmetic functions r + k n, r k n, and ρ + k n, ρ k n, which are defined, respectively, by r ± k n :={u, v Z 2 : u k ± v k =n}, ρ ± kn:={u, v Z 2 : u k ± v k =n, gcd u, v =1}. From a geometric viewpoint, these functions are associated with the starlike planar domains D ± := {u, v R 2 : 0 < u k ± v k 1}. 5 o be precise, it suffices that the curvature of D, as a function of the arclength, is twice continuously differentiable.

13 PRIMIIVE LAICE POINS 175 It is known from classic results of Krätzel [9], [10], [11] that 4.2 r ± k n =A D ± 2/k =c ± 0k 2/k + c ± 1 k 1/k 1 + O 1 k 1 k 2, with 1 n c + 0 k = 2Γ2 1 k kγ 2 k, c + 1 k =0, Γ 2 1 c k 0 k= kcos π Γ 2, k k 1 c 1 k=4ζ k 1/k 1. k 1 Our heorem readily implies provided that RH is true 1 n ρ ± k n =B D ± 2/k = 6 π 2 c± 0 k 2/k + c± 1 k 1/k 1 + O ζ k k 1 7k+1 k7k+4 +ε. Again the estimate can be improved slightly, making use of more precise representations of the error term in 4.2 see [19] Primitive Pythagorean triangles. Let us define as a primitive Pythagorean triangle any triple of natural numbers u, v, w satisfying u 2 + v 2 = w 2, u v, gcd u, v, w =1. For a large real parameter A, let pa denote the number of primitive Pythagorean triangles with area less than A. he problem to establish an asymptotic formula for pa has been attacked by Lambek and Moser [12], Wild [22], Duttlinger and Schwarz [4], Müller, Nowak and Menzer [17], and Müller and Nowak [16]. According to Lambek and Moser [12], it is known that 4.3 pa = where 1 k B D A 2 k, k=0 D := {u, v R 2 : uvu 2 v 2 < 1, 0 <v<u}. In [16] it has been shown that A D x =c 0 x+c 1 x 2/3 +O x 7/22 log x 45/22,

14 176 WERNER GEORG NOWAK with c 0 = Γ π, c 1 = 1 3 ζ 1+2 1/3. Applying the most recent version of Huxley s lattice point theorems [6], it is straightforward to sharpen this error term to O x log x 146. Using this with our heorem and 4.3, one obtains conditionally under RH pa =c 0A 1/2 +c 1A 1/3 +O A 269 +ε 1238 with c 0 = Γ π 5, ζ 1 c /3 1 = ζ /3, 269 which improves upon all earlier results of this kind. Numerically, = , while the best exponent in the error term known before was = Primitive lattice points in special asteroid-shaped domains. As a last somewhat exotic example we consider starlike sets D a := {u, v R 2 : u a + v a 1} where a is a fixed real number with 0 <a<1. It was known already to van der Corput [3] that A Da x = area D a x + 1 r< 1 a 1 2λ c r ax 1 ar/2 + O x λ, with c r a = 8 1r ζ arγ 1+ 1 a r!γ 1+ 1 a r, for λ = 1. Cf. also [18] for a generalization. Appealing again to Huxley s 3 work [6], this can be readily established for every λ> 23. hus our heorem 73 implies that if RH is true B Da x = 6 π 2 area D a x + 1 r< a c r a ζ1 ar x1 ar/2 + O x 269 +ε 619.

15 PRIMIIVE LAICE POINS 177 References [1].M. Apostol, Introduction to analytic number theory, Springer, New York - Heidelberg - Berlin, [2] R.C. Baker, he square-free divisor problem, Quart. J. Math. Oxford, , [3] J.G. van der Corput, Over roosterpunkten in het plate vlak, Dutch, hesis Groningen, [4] J. Duttlinger and W. Schwarz, Über die Verteilung der Pythagoräischen Dreiecke, Colloquium Math., , [5] D. Hensley, he number of lattice points within a contour and visible from the origin, Pacific J. Math., , [6] M.N. Huxley, Exponential sums and lattice points II, Proc. London Math. Soc., , [7] M.N. Huxley and W.G. Nowak, Primitive lattice points in convex planar domains, Acta Arithm., , [8] A. Ivić, he Riemann Zeta-function, J. Wiley & Sons, New York and Chichester, [9] E. Krätzel, Bemerkungen zu einem Gitterpunktproblem, Math. Ann., , [10], Mittlere Darstellungen natürlicher Zahlen als Differenz zweier k-ter Potenzen, Acta Arithm., , [11], Lattice Points, Dt. Verl. d. Wiss., Berlin, [12] J. Lambek and L. Moser, On the distribution of Pythagorean triangles, Pacific J. Math., , [13] E. Landau, Ausgewählte Abhandlungen zur Gitterpunktlehre, ed. by A. Walfisz, Dt. Verl. d. Wiss., Berlin, [14] H.L. Montgomery and R.C. Vaughan, he distribution of squarefree numbers, in Recent Progress in Analytic Number heory, Proc. Durham Symp. 1979, Vol. I, H. Halberstam and C. Hooley, Editors, Academic Press, London, 1981, [15] B.Z. Moroz, On the number of primitive lattice points in plane domains, Monatsh. Math., , [16] W. Müller and W.G. Nowak, Lattice points in planar domains: Applications of Huxley s Discrete Hardy-Littlewood method, in: Number theoretic analysis, Seminar Vienna , Springer Lecture Notes, 1452 eds. E. Hlawka and R. F. ichy, 1990, [17] W. Müller, W.G. Nowak and H. Menzer, On the number of primitive Pythagorean triangles, Ann. Sci. Math., Québec, , [18] W.G. Nowak, A non-convex generalization of the circle problem, J. Reine Angew. Math., , [19] W.G. Nowak, Sums and differences of two relative prime cubes, II, Proc. Czech and Slovake Number heory Conference, 1995, in press. [20] W.G. Nowak and M. Schmeier, Conditional asymptotic formulae for a class of arithmetic functions, Proc. Amer. Math. Soc., ,

16 178 WERNER GEORG NOWAK [21] K. Prachar, Primzahlverteilung, Springer Verlag, Berlin-Heidelberg-New York, [22] R.E. Wild, On the number of primitive Pythagorean triangles with area less than n, Pacific J. Math., , Received September 20, 1995 and revised December 19, his article is part of a research project supported by the Austrian Science Foundation NR. P 9892-PHY. Universität für Bodenkultur A-1180 Wien, Austria address: nowak@mail.boku.ac.at