Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant

Ulrike M. A. Vorhauer; Eduard Wirsing

Acta Arithmetica (1999)

  • Volume: 91, Issue: 1, page 43-55
  • ISSN: 0065-1036

Abstract

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1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple estimate of certain exponential integrals. In Part I we determine the order of magnitude of the Hessian as well as that of the maximum of the second derivatives for the functions h, which are third order differences of the two-dimensional Euclidean vector norm.

How to cite

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Ulrike M. A. Vorhauer, and Eduard Wirsing. "Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant." Acta Arithmetica 91.1 (1999): 43-55. <http://eudml.org/doc/207338>.

@article{UlrikeM1999,
abstract = { 1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple estimate of certain exponential integrals. In Part I we determine the order of magnitude of the Hessian as well as that of the maximum of the second derivatives for the functions h, which are third order differences of the two-dimensional Euclidean vector norm. },
author = {Ulrike M. A. Vorhauer, Eduard Wirsing},
journal = {Acta Arithmetica},
keywords = {estimates on exponential sums; lattice points in large regions; circle problem; logarithmic mean; Hessian},
language = {eng},
number = {1},
pages = {43-55},
title = {Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant},
url = {http://eudml.org/doc/207338},
volume = {91},
year = {1999},
}

TY - JOUR
AU - Ulrike M. A. Vorhauer
AU - Eduard Wirsing
TI - Three two-dimensional Weyl steps in the circle problem I. The Hessian determinant
JO - Acta Arithmetica
PY - 1999
VL - 91
IS - 1
SP - 43
EP - 55
AB - 1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple estimate of certain exponential integrals. In Part I we determine the order of magnitude of the Hessian as well as that of the maximum of the second derivatives for the functions h, which are third order differences of the two-dimensional Euclidean vector norm.
LA - eng
KW - estimates on exponential sums; lattice points in large regions; circle problem; logarithmic mean; Hessian
UR - http://eudml.org/doc/207338
ER -

References

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  1. [1] J.-R. Chen, The lattice points in a circle, Sci. Sinica 12 (1963), 633-649. 
  2. [2] L.-K. Hua, The lattice points in a circle, Quart. J. Math. Oxford Ser. 13 (1942), 18-29. Zbl0060.11605
  3. [3] V. Jarník, Zentralblatt für Mathematik und ihre Grenzgebiete 10 (1935), 104. 
  4. [4] E. Krätzel, Lattice Points, Kluwer Acad. Publ., Dordrecht, 1988. Zbl0675.10031
  5. [5] S.-H. Min, On the order of ζ(1/2 + it), Trans. Amer. Math. Soc. 65 (1949), 448-472. Zbl0033.26901
  6. [6] H.-E. Richert, Verschärfung der Abschätzung beim Dirichletschen Teilerproblem, Math. Z. 58 (1953), 204-218. Zbl0050.04202
  7. [7] E. C. Titchmarsh, On Epstein's zeta-function, Proc. London Math. Soc. (2) 36 (1934), 485-500. Zbl0008.30101
  8. [8] E. C. Titchmarsh, The lattice-points in a circle, Proc. London Math. Soc. (2) 38 (1935), 96-115; Corrigendum, Proc. London Math. Soc., 555. Zbl0010.10403
  9. [9] E. C. Titchmarsh, On the order of ζ(1/2 + it), Quart. J. Math. Oxford Ser. 13 (1942), 11-17. Zbl0061.08301
  10. [10] A. Walfisz, Zentralblatt für Mathematik und ihre Grenzgebiete 8 (1934), 301. 

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