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Integral Equation for the Number of Integer Points in a Circle ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS { N. 41{2019 (522{525) 522 INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE Magomed A. Chakhkiev Gagik S. Sulyan Manya A. Ziroyan Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation Nikolay P. Tretyakov Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation and School of Public Policy The Russian Presidential Academy of National Economy and Public Administration Prospect Vernadskogo, 84, Moscow 119571 Russian Federation Saif A. Mouhammad∗ Department of Physics Faculty of Science Taif University Taif, AL-Haweiah Kingdom of Saudi Arabia [email protected] Abstract. The problem is to obtain the most accurate upper estimate for the absolute value of the difference between the number of integer points in a circle and its area (when the radius tends to infinity). In this paper we obtain an integral equation for the function expressing the dependence of the number of integer points in a circle on its radius. The kernel of the equation contains the Bessel functions of the first kind, and the equation itself is a kind of the Hankel transform. Keywords: Gauss circle problem, integral equation, Hankel transform. 1. The problem and calculations The Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin andp with given radius. Let us consider the circle K(R): x2 + y2 ≤ R and let A( R) be the number of ∗. Corresponding author INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE 523 p points with integer coordinates within this circle. As R increases, A( R) is approximatelyp equal to the area inside the circle πR . Let us define ∆(R) = jA( R)−πRj. The Gauss circle problem consists in estimating the upper bound for ∆(R) when R ! 1 as much precisely as possible. Similarly, the Dirichlet divisor problem consists in finding the number of integer points under the hyperbola K1(R): xy ≤ R; 0 < x ≤ R; 0 < y ≤ R. The problems of Gauss and Dirichlet were investigated by many authors. Gauss himself found the estimation O(R1=2) for∆(R). Voronoy [1] obtained the result O(R1=3lnR) for the Dirichlet problem, while Serpinksij found the estimation O(R1=3) in 1903 and Hua O(R13=40) in 1942 [2] for the circle. Hardy and Littlewood [3] proved that it is impossible to get a better estimation than O(R1=4ln2R). Probably the most precise estimation for the circle up to now is O(R131=208) obtained by Huxley in the early 2000s [4]. To give the readers more information about the problem, they can take a look to the references [5, 6]. Our aim is to derive an integral equation for the number of integer points in a circle. P 1 sin 2πnϵ Let δϵ(x) = 1 + n=1 2πnϵ cos 2πnx be a periodic function with period 1, which is tending to the periodic delta-function of Dirac when ϵ ! 0. Let us consider the integral p Z Z (1) Aϵ( R) = δϵ(x)δϵ(y) dx dy: K (R) p p We can see that limϵ!0 Aϵ( R) = A( R). Let us calculate the integral in the Pright hand side of (1). The function δϵ(x) may be written as follows: δϵ(x) = 1 2πinx sin 2πnϵ 6 −∞ cne , where cn = 2πnϵ for n = 0 and cn = 1 for n = 0. Then p Z Z Aϵ( R) = δϵ(x)δϵ(y) dx dy K (R) Z p Z R 2π X+1 2πir (n cos ϕ+m sin ϕ) = r dr cncme dϕ. 0 0 n;m=−∞ After changing the order of integration and summation and transforming the function n cos ϕ + m sin ϕ, the expression takes the form: Z p Z p X+1 R 2π p 2 2 2πir n +m cos(ϕ+ϕ0) Aϵ( R) = cncm r dr e dϕ n;m=−∞ 0 0 Z p Z X+1 R 2π p 2πir n2+m2cos(ϕ) = cncm r dr e dϕ. n;m=−∞ 0 0 R 1 π ix cos ϕ Taking into account that J0(x) = π 0 e dϕ is the Bessel function of the first kind and zero order, we get: Z p p X+1 R p 2 2 (2) Aϵ( R) = 2π cncm r J0(2πr n + m ) dr: n;m=−∞ 0 524M.A. CHAKHKIEV, G.S. SULYAN, M.A. ZIROYAN, N.P. TRETYAKOV, S.A. MOUHAMMAD Using the relation d (3) xJ (x) = (xJ (x)) ; 0 dx 1 where J1(x) is the Bessel function of the first kind, we get by integrating from (2): 1 p p p X+ 2 2 J1(2π R(n + m )) (4) Aϵ( R) = R cncm p : 2 2 n;m=−∞ n + m Since Cn tend to 1 while ϵ ! 0 we get: 1 p p p X+ 2 2 J1(2π R(n + m )) Aϵ( R) = R p 2 2 −∞ n + m n;m= p p X J (2π R(n2 + m2)) = πR + R 1 p : n2 + m2 n2+m2=06 p The series for A( R) does not converge absolutely, but it sums up to the number of integer points in a round area with radius R1=2 (if R is an integer, the points onp the circumference are counted with the coefficient 1=2). The expression A( R) may be rewritten in the following way: Z Z p p p p 1 1 2 2 J1(2π R x + y ) A( R) = lim R p δϵ(x) δϵ(y)dxdy ϵ!0 −∞ −∞ x2 + y2 Z Z p 1 p 2π = lim R J (2πr R) dr δ (r cos ϕ) δ (r sin ϕ) dϕ. ! 1 ϵ ϵ ϵ 0 0 0 Since (see above) Z 2π X+1 p 2 2 δϵ(r cos ϕ) δϵ(r sin ϕ) dϕ = 2π cncm J0(2πr n + m ); 0 n;m=−∞ we get: Z p p +1 p X+1 p 2 2 A( R) = lim 2π R J1(2πr R) cncm J0(2πr n + m ) dr: ϵ!0 0 n;m=−∞ p Let us denote R = ρ. Using the property (3) of Bessel functions and integrat- ing by parts, we get: ! Z ( ) 1 p +1 X+ 2 2 1 − 0 J1(2pπr n + m ) A(ρ)=lim J1(2πρr) 2πρJ1(2πρr) cncm dr: ϵ!0 r 2 2 0 n;m=−∞ n + m INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE 525 In view of (4), the following expression is derived: Z ( ) +1 1 2πρ A(ρ) = lim J (2πρr) − J 0 (2πρr) A (r) dr: ! 2 1 1 ϵ ϵ 0 0 r r 0 ! Now, replacing J1(2πρr) according to formula (3) and taking the limit ϵ 0 we finally obtain the integral equation: Z +1 (5) A(ρ) = A(r) K(ρ, r) dr; 0 where the core is 2 2πρ K(ρ, r) = J (2πρr) − J (2πρr): r2 1 r 0 2. Conclusion R +1 Let us note that the integral transform F (ρ) = 0 A(r) K(ρ, r) dr with the core K(ρ, r) = r Js(2πρr); s ≥ −1=2 is known as the Hankel transform. Thus, for the function expressing the dependence of the number of integer points in a circle on its radius, the integral equation is obtained, which is a kind of the Hankel transform. It can be used for further investigations of the Gauss circle problem. Also of interest is a possible generalization of the methodology and applying it to other similar problems (for example, the Dirichlet divisor problem). References [1] G. Voronoi, Sur un probleme du calcul dec fonctions asymptoliques, J. fur Math., 126 (1903), 241-282. [2] L. K. Hua, The Lattice points in a circle, Quart. J. Math., Oxford, 13 (1942), 18-29. [3] G.H. Hardy and J.Littlewood, The trigonometrical series associated with the elliptic function, Acta Math., 37 (1914), 193-239. [4] M. N. Huxley, Exponential sums and lattice points. III, Proc. London Math. Soc., (3) 87 (2003), 591-609. [5] M.N. Huxley, Integer points, exponential sums and the Riemann zeta func- tion, Number theory for the millennium, II (Urbana, IL, 2000), 275-290, A. K. Peters, Natick, MA, 2002. [6] D. Hilbert and S. Cohen-Vossen, Geometry and the imagination, New York, Chelsea, 37-38, 1999. Accepted: 10.07.2018.
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