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  • 1 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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Abstract

We improve the upper bound for the lattice point discrepancy of large spheres under conjectural properties of the real L-functions. In connection with this we give some new unconditional estimates for exponential and character sums of independent interest.

  • [1] Bykovskiî, V. A. 1997 Density theorems and the mean value of arithmetic functions on short intervals J. Math. Sci. 83 720730 . (Russian) translation in.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [2] Chamizo, F. Cristóbal, E. Ubis, A. 2007 Visible points in the sphere J. Number Theory 126 200211 .

  • [3] Chamizo, F. Cristóbal, E. Ubis, A. 2009 Lattice points in rational ellipsoids J. Math. Anal. Appl. 350 283289 .

  • [4] Chamizo, F. Iwaniec, H. 1995 On the sphere problem Rev. Mat. Iberoamericana 11 417429 .

  • [5] Chen, J.-R. 1963 Improvement on the asymptotic formulas for the lattice points in a region of the three dimensions (II) Sci. Sinica 12 751764.

    • Search Google Scholar
    • Export Citation
  • [6] Ellison, W. J., Les nombres premiers, en collaboration avec Michel Mendès France, Hermann (Paris, 1975).

  • [7] Gauss, C. F. 1986 Disquisitiones Arithmeticae Springer-Verlag New York.

  • [8] Graham, S. W. Kolesnik, G. 1991 Van der Corput's Method of Exponential Sums London Mathematical Society Lecture Note Series 126 Cambridge University Press Cambridge .

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [9] Grosswald, E. 1985 Representation of Integers as Sums of Squares Springer Verlag New York .

  • [10] Heath-Brown, D. R. 1995 A mean value estimate for real character sums Acta Arith. 72 235275.

  • [11] Heath-Brown, D. R. 1999 Lattice points in the sphere Number Theory in Progress 2 Gruyter Berlin 883892 Zakopane–Kościelisko, 1997.

    • Search Google Scholar
    • Export Citation
  • [12] Huxley, M. N. 2005 Exponential sums and the Riemann zeta function V Proc. London Math. Soc. 90 141 .

  • [13] Ivić, A. Krätzel, E. Kühleitner, M. Nowak, W. G. 2006 Lattice points in large regions and related arithmetic functions: recent developments in a very classic topic, Elementare und analytische Zahlentheorie Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main 20 89128.

    • Search Google Scholar
    • Export Citation
  • [14] Iwaniec, H. Kowalski, E. 2004 Analytic Number Theory AMS Coll. Publ. 53 Providence.

  • [15] Krätzel, E. 1988 Lattice Points, Mathematics and its Applications 33 Kluwer Academic Publishers Group Dordrecht.

  • [16] Krätzel, E. Nowak, W. G. 2008 The lattice discrepancy of bodies bounded by a rotating Lamé's curve Monatsh. Math. 154 145156 .

  • [17] Landau, E. 1924 Über Gitterpunkte in mehrdimensionalen Ellipsoiden Math. Zeitschrift 21 126132 .

  • [18] Montgomery, H. L. 1994 Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis CBMS Regional Conference Series in Mathematics 84 AMS Providence.

    • Search Google Scholar
    • Export Citation
  • [19] Nowak, W. G. 1986 On the lattice rest of a convex body in ℝs, II Arch. Math., (Basel) 47 232237 .

  • [20] Nowak, W. G. 2008 The lattice point discrepancy of a torus in ℝ3 Acta Math. Hungar. 120 179192 .

  • [21] Nowak, W. G. 2009 On the lattice discrepancy of ellipsoids of rotation Unif. Distrib. Theory 4 101114.

  • [22] Patterson, S. J. 1988 An introduction to the Theory of the Riemann Zeta-function Cambridge University Press Cambridge.

  • [23] Phillips, E. 1993 The zeta-function of Riemann; further developments of Van der Corput's method Q.J. Math. 4 209225 .

  • [24] Szegö, G. 1926 Beiträge zur Theorie de Laguerreschen Polynome, II. Zahlentheoretische Anwendungen Math. Z. 25 388404 .

  • [25] Tsang, K. M. 2000 Counting lattice points in the sphere Bull. London Math. Soc. 32 679688 .

  • [26] Vinogradov, I. M. 1963 On the number of integer points in a sphere Izv. Akad. Nauk SSSR Ser. Mat. 27 957968.

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  • SJR Quartile Score (2019): Q2 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.538
  • Scimago Journal Rank (2018): 0.488
  • SJR Hirsch-Index (2018): 36
  • SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder Magyar Tudományos Akadémia
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Springer Nature Switzerland AG
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ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)