American Institute of Mathematical Sciences

January  2016, 15(1): 1-8. doi: 10.3934/cpaa.2016.15.1

Average number of lattice points in a disk

 1 3633 19th St., San Francisco, CA 94110, United States 2 Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States

Received  November 2014 Revised  October 2015 Published  December 2015

The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.
Citation: Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure and Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1
References:
 [1] M. Begué, T. Kalloniatis and R. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 13500023 (32 pages). doi: 10.1142/S0218348X13500023. [2] P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math J., 67, 461-481. doi: 10.1215/S0012-7094-92-06718-4. [3] P. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems, Duke Math J., 70, 655-682. doi: 10.1215/S0012-7094-93-07015-9. [4] M. Huxley, The mean lattice discrepancy, Proc. Edinburg Math. Soc., 38 (1995), 523-531. doi: 10.1017/S0013091500019313. [5] H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloq. Publ. vol 53, 2004. [6] S. Jayakar and R. Strichartz, Average number of lattice points in a disk, http://www.math.cornell.edu/ sujay/lattice, 2012. [7] N. Lebedev, Special Functions and Their Applications, Dover Publications, New York, 1965. [8] W. Müller, On the average order of the lattice rest of a convex body, Acta Arith., 80 (1997), 89-100. [9] C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, Princeton Univ. Press, Princeton, 2014. doi: 10.1515/9781400850549. [10] E. Stein and R. Shakarchi, Functional Analysis, Princeton Univ. Press, 2011. [11] R. Strichartz, Average error for spectral asymptotics on surfaces, Comm. Pure Appl. Analysis, 15 (2016), 9-39.

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References:
 [1] M. Begué, T. Kalloniatis and R. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 13500023 (32 pages). doi: 10.1142/S0218348X13500023. [2] P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals, Duke Math J., 67, 461-481. doi: 10.1215/S0012-7094-92-06718-4. [3] P. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems, Duke Math J., 70, 655-682. doi: 10.1215/S0012-7094-93-07015-9. [4] M. Huxley, The mean lattice discrepancy, Proc. Edinburg Math. Soc., 38 (1995), 523-531. doi: 10.1017/S0013091500019313. [5] H. Iwaniec and E. Kowalski, Analytic Number Theory, AMS Colloq. Publ. vol 53, 2004. [6] S. Jayakar and R. Strichartz, Average number of lattice points in a disk, http://www.math.cornell.edu/ sujay/lattice, 2012. [7] N. Lebedev, Special Functions and Their Applications, Dover Publications, New York, 1965. [8] W. Müller, On the average order of the lattice rest of a convex body, Acta Arith., 80 (1997), 89-100. [9] C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian, Princeton Univ. Press, Princeton, 2014. doi: 10.1515/9781400850549. [10] E. Stein and R. Shakarchi, Functional Analysis, Princeton Univ. Press, 2011. [11] R. Strichartz, Average error for spectral asymptotics on surfaces, Comm. Pure Appl. Analysis, 15 (2016), 9-39.
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