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On a generalized divisor problem I

Published online by Cambridge University Press:  22 January 2016

Yuk-Kam Lau
Yuk-Kam Lau*
Affiliation:
Institut Élie Cartan, Université Henri Poincaré (Nancy 1), 54506 Vandoeuvre lés Nancy Cedex, France
*
Department of Mathematics, The University of Hong Kong, Pokfulam Road, HONG KONG, [email protected]
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Abstract

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We give a discussion on the properties of Δa(x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a < 0.

Keywords

11N64
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 165 , March 2002 , pp. 71 - 78
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[1] Chowla, S., Contributions to the analytic theory of numbers, Math. Z., 35 (1932), 279–299.Google Scholar
[2] Hafner, J.L., On the Average Order of a Class of Arithmetical Functions, J. Number Theory, 15 (1982), 36–76.CrossRefGoogle Scholar
[3] Heath-Brown, D.R. and Tsang, K., Sign Changes of E(T), Δ(x) and P(x), J. Number Theory, 49 (1984), 73–83.Google Scholar
[4] Ivić, A., Large values of certain number-theoretic error terms, Acta Arith., 56 (1990), 135–159.Google Scholar
[5] Jutila, M., On the divisor problem for short intervals, Ann. Univ. Turkuensis Ser. A, I 186 (1984), 23–30.Google Scholar
[6] Kiuchi, I. and Tanigawa, Y., The mean value theorem of the divisor problem for short intervals, Arch. Math. (Basel), 71 (1998), 445–453.Google Scholar
[7] Lam, K.-Y. and Tsang, K.-M., The Mean Square of the Error Term in a Generalization of the Dirichlet Divisor Problem, Analytic Number Theory, edited by Y.Motohashi, Cambridge University Press, 1997.Google Scholar
[8] Lau, Y.-K., On a generalized divisor problem II, manuscript.Google Scholar
[9] Meurman, T., The mean square of the error term in a generalization of Dirichlet’s divisor problem, Acta Arith., 74 (1996), 351–361.Google Scholar
[10] Pétermann, Y.-F.S., About a Theorem of Paolo Codecàs and Omega Estimates for Arithmetical Convolutions, Second Part, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17 (1990), 343–353.Google Scholar