Circle problem and the spectrum of the Laplace operator on closed 2-manifolds

D. A. Popov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 74 (2019) no. 5, pp. 909-925

Résumé

In this survey the circle problem is treated in the broad sense, as the problem of the asymptotic properties of the quantity $P(x)$, the remainder term in the circle problem. A survey of recent results in this direction is presented. The main focus is on the behaviour of $P(x)$ on short intervals. Several conjectures on the local behaviour of $P(x)$ which lead to a solution of the circle problem are presented. A strong universality conjecture is stated which links the behaviour of $P(x)$ with the behaviour of the second term in Weyl's formula for the Laplace operator on a closed Riemannian 2-manifold with integrable geodesic flow. Bibliography: 43 titles.

Détail

Keywords: circle problem, short intervals, quantum chaos, universality conjecture.
Mots-clés : Voronoi's formula

Bibliographie

[1] Kai-Man Tsang, “Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function”, Sci. China Math., 53:9 (2010), 2561–2572 | DOI | MR | Zbl

[2] A. Bäcker, F. Steiner, “Quantum chaos and quantum ergodicity”, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, 717–751 | DOI | MR | Zbl

[3] P. Sarnak, “Arithmetic quantum chaos”, The Shur lectures (1992), Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat-Gan, 1995, 183–236 | MR | Zbl

[4] A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin, 1993, xiv+222 pp. | DOI | MR | MR | Zbl | Zbl

[5] S. W. Graham, G. Kolesnik, Van der Corput's method of exponential sums, London Math. Soc. Lecture Note Ser., 126, Cambridge Univ. Press, Cambridge, 1991, vi+120 pp. | DOI | MR | Zbl

[6] G. Voronoï, “Sur le développement, à l'aide des fonctions cylindriques, des sommes doubles $\sum f(pm^2+2qmn+rn^2)$, où $pm^2+2qmn+rn^2$ est une forme positive à coefficients entiers”, Verhandlungen des dritten internationalen Mathematiker-Kongresses (Heidelberg, 1904), Teubner, Leipzig, 1905, 241–245 | Zbl

[7] G. H. Hardy, “On the expression of a number as the sum of two squares”, Quart. J. Pure Appl. Math., 46 (1915), 263–283 | Zbl

[8] E. Landau, Vorlesungen über Zahlentheorie, v. 2, Hirzel, Leipzig, 1927, vii+308 pp. | MR | Zbl

[9] G. H. Hardy, M. Reisz, The general theory of Dirichlet's series, Cambridge Tracts in Math. and Math. Phys., 18, Cambridge Univ. Press, Cambridge, 1915, 78 pp. | MR | Zbl

[10] E. Bombieri, H. Iwaniec, “On the order of $\zeta(\frac{1}{2}+it)$”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13:3 (1986), 449–472 | MR | Zbl

[11] H. Iwaniec, C. J. Mozzochi, “On the divisor and circle problems”, J. Number Theory, 29:1 (1988), 60–93 | DOI | MR | Zbl

[12] M. N. Huxley, Area, lattice points, and exponential sums, London Math. Soc. Monogr. (N. S.), 13, The Clarendon Press, Oxford Univ. Press, New York, 1996, xii+494 pp. | MR | Zbl

[13] M. N. Huxley, “Exponential sums and lattice points. II”, Proc. London Math. Soc. (3), 66:2 (1993), 279–301 | DOI | MR | Zbl

[14] M. N. Huxley, “Exponential sums and lattice points. III”, Proc. London Math. Soc. (3), 87:3 (2003), 591–609 | DOI | MR | Zbl

[15] J. Bourgain, N. Watt, Mean square of zeta function, circle problem and divisor problem revisited, 2017, 23 pp., arXiv: 1709.04340

[16] K. S. Gangadharan, “Two classical lattice point problems”, Proc. Cambridge Philos. Soc., 57:4 (1961), 699–721 | DOI | MR | Zbl

[17] J. L. Hafner, “New omega theorems for two classical lattice point problems”, Invent. Math., 63:2 (1981), 181–186 | DOI | MR | Zbl

[18] K. Soundararajan, “Omega results for the divisor and circle problems”, Int. Math. Res. Not., 2003:36 (2003), 1987–1998 | DOI | MR | Zbl

[19] A. Ivić, “Large values of the error term in the divisor problem”, Invent. Math., 71:3 (1983), 513–520 | DOI | MR | Zbl

[20] D. R. Heath-Brown, “The distribution and moments of the error term in the Dirichlet divisor problem”, Acta Arith., 60:4 (1992), 389–415 | DOI | MR | Zbl

[21] E. Preissmann, “Sur la moyenne quadratique du terme de reste du problème du cercle”, C. R. Acad. Sci. Paris Sér. I Math., 306:4 (1988), 151–154 | DOI | MR | Zbl

[22] W. G. Nowak, “Lattice points in a circle: an improved mean-square asymptotics”, Acta Arith., 113:3 (2004), 259–272 | DOI | MR | Zbl

[23] Kai-Man Tsang, “Higher-power moments of $\Delta(x)$, $E(t)$ and $P(x)$”, Proc. London Math. Soc. (3), 65:1 (1992), 65–84 | DOI | MR | Zbl

[24] Wenguang Zhai, “On higher-power moments of $\Delta(x)$. II”, Acta Arith., 114:1 (2004), 35–54 | DOI | MR | Zbl

[25] W. G. Nowak, “On the divisor problem: moments of $\Delta(x)$ over short intervals”, Acta Arith., 109:4 (2003), 329–341 | DOI | MR | Zbl

[26] Yuk-Kam Lau, Kai-Man Tsang, “Moments over short intervals”, Arch. Math. (Basel), 84:3 (2005), 249–257 | DOI | MR | Zbl

[27] A. Ivić, P. Sargos, “On the higher moments of the error term in divisor problem”, Illinois J. Math., 51:2 (2007), 353–377 | DOI | MR | Zbl

[28] D. A. Popov, “Bounds and behaviour of the quantities $P(x)$, $\Delta(x)$ on short intervals”, Izv. Math., 80:6 (2016), 1213–1230 | DOI | DOI | MR | Zbl

[29] M. Jutila, “On the divisor problem for short intervals”, Ann. Univ. Turku. Ser. A I, 186 (1984), 23–30 | MR | Zbl

[30] A. Ivić, “On the divisor function and the Riemann zeta-function in short intervals”, Ramanujan J., 19:2 (2009), 207–224 | DOI | MR | Zbl

[31] D. R. Heath-Brown, K.-M. Tsang, “Sign changes of $E(T)$, $\Delta(x)$ and $P(x)$”, J. Number Theory, 49 (1994), 73–83 | DOI | MR | Zbl

[32] A. Ivić, Wenguang Zhai, “On the Dirichlet divisor problem in short intervals”, Ramanujan J., 33:3 (2014), 447–465 | DOI | MR | Zbl

[33] P. M. Bleher, Zheming Cheng, F. J. Dyson, J. L. Lebowitz, “Distribution of the error term for the number of lattice points inside a shifted circle”, Comm. Math. Phys., 154:3 (1993), 433–469 | DOI | MR | Zbl

[34] Yuk-Kam Lau, “On the tails of the limiting distribution function of the error term in the Dirichlet divisor problem”, Acta Arith., 100:4 (2001), 329–337 | DOI | MR | Zbl

[35] J. Steinig, “The changes of sign of certain arithmetical error-terms”, Comment. Math. Helv., 44 (1969), 385–400 | DOI | MR | Zbl

[36] A. Ivić, “Large values of certain number-theoretic error term”, Acta Arith., 56:2 (1990), 135–159 | DOI | MR | Zbl

[37] V. I. Arnold, A. Avez, Problèmes ergodiques de la mécanique classique, Monographies Internationales de Mathématiques Modernes, 9, Gauthier-Villars, Paris, 1967, ii+243 pp. | MR | Zbl | Zbl

[38] M. L. Mehta, Random matrices, Pure Appl. Math. (Amst.), 142, 3rd ed., Elsevier/Academic Press, Amsterdam, 2004, xviii+688 pp. | MR | Zbl

[39] P. M. Bleher, “Distribution of energy levels of a quantum free particle on a surface of revolution”, Duke Math. J., 74:1 (1994), 45–93 | DOI | MR | Zbl

[40] D. V. Kosygin, A. A. Minasov, Ya. G. Sinai, “Statistical properties of the spectra of Laplace–Beltrami operators on Liouville surfaces”, Russian Math. Surveys, 48:4 (1993), 1–142 | DOI | MR | Zbl

[41] P. M. Bleher, D. V. Kosygin, Ya. G. Sinai, “Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae”, Comm. Math. Phys., 170:2 (1995), 375–403 | DOI | MR | Zbl

[42] D. A. Popov, “On the second term in the Weyl formula for the spectrum of the Laplace operator on the two-dimensional torus and the number of integer points in spectral domains”, Izv. Math., 75:5 (2011), 1007–1045 | DOI | DOI | MR | Zbl

[43] A. B. Venkov, “Spectral theory of automorphic functions, the Selberg zeta-function, and some problems of analytic number theory and mathematical physics”, Russian Math. Surveys, 34:3 (1979), 79–153 | DOI | MR | Zbl