SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

Department of Mathematics, Rutgers University Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA; alexander.walker@rutgers.edu

Corresponding

E-mail address:

thomas.hulse@bc.edu ;

kuanchi3@mail.sysu.edu.cn ;

d.lowry@warwick.ac.uk ;

alexander.walker@rutgers.edu

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Abstract

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$ , where $P_{k}(n)$ is the discrepancy between the volume of the $k$ -dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$ , in dimensions $k\geqslant 3$ . We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$ , along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$ . This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

MSC classification

Primary: 11N37: Asymptotic results on arithmetic functions

Secondary: 11F30: Fourier coefficients of automorphic forms

Type

Research Article

Information

Forum of Mathematics, Sigma , Volume 6 , 2018 , e24

DOI: https://doi.org/10.1017/fms.2018.26

Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

Bourgain, J. and Watt, N., ‘Mean square of zeta function, circle problem and divisor problem revisited’. arXiv preprint, arXiv:1709.04340 (2017).Google Scholar

Choi, S. K. K., Kumchev, A. V. and Osburn, R., ‘On sums of three squares’, Int. J. Number Theory 1(2) (2005), 161–173.CrossRef Google Scholar

Deshouillers, J.-M. and Iwaniec, H., ‘Kloosterman sums and Fourier coefficients of cusp forms’, Invent. Math. 70(2) (1982/83), 219–288.CrossRef Google Scholar

Goldfeld, D., Automorphic Forms and L-functions for the Group GL(n, R), Cambridge Studies in Advanced Mathematics, 99 (Cambridge University Press, Cambridge, 2015), With an appendix by Kevin A. Broughan, Paperback edition of the 2006 original.Google Scholar

Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series, and Products, 8th edn (Elsevier/Academic Press, Amsterdam, 2015), Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition.Google Scholar

Gupta, S. D., ‘Corrigendum to my paper ‘The Rankin–Selberg method on congruence subgroups’’, Illinois J. Math. 44(4) (2000), 924–926.Google Scholar

Gupta, S. D., ‘The Rankin–Selberg method on congruence subgroups’, Illinois J. Math. 44(1) (2000), 95–103.Google Scholar

Heath-Brown, D. R., ‘Lattice points in the sphere’, inNumber Theory in Progress, Vol. 2 (Zakopane-Kościelisko 1997) (de Gruyter, Berlin, 1999), 883–892.Google Scholar

Hoffstein, J. and Hulse, T. A., ‘Multiple Dirichlet series and shifted convolutions’, J. Number Theory 161 (2016), 457–533. With an appendix by Andre Reznikov.CrossRef Google Scholar

Hulse, T. A., Kuan, C. I., Lowry-Duda, D. and Walker, A., ‘The Laplace transform of the second moment in the Gauss circle problem’. arXiv preprint, arXiv:1705.04771 (2017).Google Scholar

Hulse, T. A., Kuan, C. I., Lowry-Duda, D. and Walker, A., ‘The second moment of sums of coefficients of cusp forms’, J. Number Theory 173 (2017), 304–331.CrossRef Google Scholar

Hulse, T. A., Kuan, C. I., Lowry-Duda, D. and Walker, A., ‘Short-interval averages of sums of Fourier coefficients of cusp forms’, J. Number Theory 173 (2017), 394–415.CrossRef Google Scholar

Huxley, M. N., ‘Introduction to Kloostermania’, inElementary and Analytic Theory of Numbers (Warsaw, 1982), Banach Center Publ., 17 (PWN, Warsaw, 1985), 217–306.Google Scholar

Huxley, M. N., ‘Exponential sums and lattice points. III’, Proc. Lond. Math. Soc. (3) 87(3) (2003), 591–609.CrossRef Google Scholar

Ivić, A., ‘A note on the Laplace transform of the square in the circle problem’, Studia Sci. Math. Hungar. 37(3–4) (2001), 391–399.Google Scholar

Ivić, A., Krätzel, E., Kühleitner, M. and Nowak, W. G., ‘Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic’, inElementare und analytische Zahlentheorie, Publications of the Scientific Society at the Johann Wolfgang Goethe University, Frankfurt am Main. Franz Steiner Verlag Stuttgart, Postfach 10 10 61, 70009 Stuttgard, Germany, 20 (Franz Steiner Verlag Stuttgart, Stuttgart, 2006), 89–128.Google Scholar

Iwaniec, H., Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, 53 (American Mathematical Society, Providence, RI, 2002), Revista Matemática Iberoamericana, Madrid, second edition.CrossRef Google Scholar

Iwaniec, H. and Kowalski, E., Analytic Number Theory, American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar

Jarník, V., ‘Über die Mittelwertsätze der Gitterpunktlehre. V’, Časopis Pěst. Mat. Fys. 69 (1940), 148–174.Google Scholar

Kıral, E. M., ‘Subconvexity for half integral weight L-functions’, Math. Z. 281(3–4) (2015), 689–722.CrossRef Google Scholar

Krätzel, E., Analytische Funktionen in der Zahlentheorie, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 139 (B. G. Teubner, Stuttgart, 2000).CrossRef Google Scholar

Kubota, T., Elementary Theory of Eisenstein Series (Kodansha Ltd., Tokyo; Halsted Press [John Wiley & Sons], New York–London–Sydney, 1973).Google Scholar

Landau, E., ‘Über die Gitterpunkte in einem Kreise’, Math. Z. 5(3–4) (1919), 319–320.CrossRef Google Scholar

Landau, E., Vorlesungen über Zahlentheorie. zweiter Band (Chelsea Publishing Co., New York, 1969).Google Scholar

Lau, Y.-K., ‘On the mean square formula of the error term for a class of arithmetical functions’, Monatsh. Math. 128(2) (1999), 111–129.CrossRef Google Scholar

Lau, Y.-K. and Tsang, K.-M., ‘On the mean square formula of the error term in the Dirichlet divisor problem’, Math. Proc. Cambridge Philos. Soc. 146(2) (2009), 277–287.CrossRef Google Scholar

Montgomery, H. L. and Vaughan, R. C., Multiplicative Number Theory I: Classical Theory, Vol. 97 (Cambridge University Press, Cambridge, 2006).CrossRef Google Scholar

Müller, W., ‘The mean square of Dirichlet series associated with automorphic forms’, Monatsh. Math. 113(2) (1992), 121–159.CrossRef Google Scholar

Sierpiński, W., ‘O pewnem zagadnieniu z rachunku funckcyi asymptotycnych’, Prace mat.-fiz (1906), 77–118.Google Scholar

Vinogradov, I. M., ‘On the number of integer points in a sphere’, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 957–968.Google Scholar

Zagier, D., ‘The Rankin–Selberg method for automorphic functions which are not of rapid decay’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3) (1982), 415–437. 1981.Google Scholar

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SECOND MOMENTS IN THE GENERALIZED GAUSS CIRCLE PROBLEM

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