# 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

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topUlrike 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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- [5] S.-H. Min, On the order of ζ(1/2 + it), Trans. Amer. Math. Soc. 65 (1949), 448-472. Zbl0033.26901
- [6] H.-E. Richert, Verschärfung der Abschätzung beim Dirichletschen Teilerproblem, Math. Z. 58 (1953), 204-218. Zbl0050.04202
- [7] E. C. Titchmarsh, On Epstein's zeta-function, Proc. London Math. Soc. (2) 36 (1934), 485-500. Zbl0008.30101
- [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] E. C. Titchmarsh, On the order of ζ(1/2 + it), Quart. J. Math. Oxford Ser. 13 (1942), 11-17. Zbl0061.08301
- [10] A. Walfisz, Zentralblatt für Mathematik und ihre Grenzgebiete 8 (1934), 301.

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