Average number of lattice points in a disk

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

Average number of lattice points in a disk

Sujay Jayakar ^1, and Robert S. Strichartz ^2,

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

Abstract
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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.

Keywords: Bessel function., Lattice points, Weyl asymptotics.

Mathematics Subject Classification: Primary: 35J05; Secondary: 42B9.

Citation: Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure & 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,, \emph{Fractals}, 21 (2013). doi: 10.1142/S0218348X13500023.
[2]	P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals,, \emph{Duke Math J.}, (): 461. 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,, \emph{Duke Math J.}, (): 655. doi: 10.1215/S0012-7094-93-07015-9.
[4]	M. Huxley, The mean lattice discrepancy,, \emph{Proc. Edinburg Math. Soc.}, 38 (1995), 523. 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,, \url{http://www.math.cornell.edu/ sujay/lattice}, (2012).
[7]	N. Lebedev, Special Functions and Their Applications,, Dover Publications, (1965).
[8]	W. Müller, On the average order of the lattice rest of a convex body,, \emph{Acta Arith.}, 80 (1997), 89.
[9]	C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (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,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 9.

show all references

References:

[1]	M. Begué, T. Kalloniatis and R. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet,, \emph{Fractals}, 21 (2013). doi: 10.1142/S0218348X13500023.
[2]	P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals,, \emph{Duke Math J.}, (): 461. 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,, \emph{Duke Math J.}, (): 655. doi: 10.1215/S0012-7094-93-07015-9.
[4]	M. Huxley, The mean lattice discrepancy,, \emph{Proc. Edinburg Math. Soc.}, 38 (1995), 523. 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,, \url{http://www.math.cornell.edu/ sujay/lattice}, (2012).
[7]	N. Lebedev, Special Functions and Their Applications,, Dover Publications, (1965).
[8]	W. Müller, On the average order of the lattice rest of a convex body,, \emph{Acta Arith.}, 80 (1997), 89.
[9]	C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (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,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 9.

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