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Three two-dimensional Weyl steps in the circle problem II. The logarithmic Riesz mean for a class of arithmetic functions

100%

Vorhauer U.

Acta Arithmetica

1999

tom 91

nr 1

57-73

1. Summary. In Part II we study arithmetic functions whose Dirichlet series satisfy a rather general type of functional equation. For the logarithmic Riesz mean of these functions we give a representation involving finite trigonometric sums. An essential tool here is the saddle point method. Estimation of the exponential sums in the special case of the circle problem will be the topic of Part III.

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

63%

Vorhauer U., Wirsing E.

Acta Arithmetica

1999

tom 91

nr 1

43-55

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.

Ograniczanie wyników

2 Acta Arithmetica

2 Vorhauer U.

1 Wirsing E.

2 1999

Wyniki wyszukiwania

Three two-dimensional Weyl steps in the circle problem II. The logarithmic Riesz mean for a class of arithmetic functions

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