EMIS ELibM Electronic Journals Publications de l'Institut Mathématique, Nouvelle Série
Vol. 92(106), pp. 97–108 (2012)

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ON A RELATION BETWEEN SUMS OF ARITHMETICAL FUNCTIONS AND DIRICHLET SERIES

Hideaki Ishikawa and Yuichi Kamiya

Faculty of Education, Nagasaki University, Nagasaki, Japan and Department of Modern Economics, Faculty of Economics, Daito Bunka University, Saitama, Japan

Abstract: We introduce a concept called good oscillation. A function is called good oscillation, if its $m$-tuple integrals are bounded by functions having mild orders. We prove that if the error terms coming from summatory functions of arithmetical functions are good oscillation, then the Dirichlet series associated with those arithmetical functions can be continued analytically over the whole plane. We also study a sort of converse assertion that if the Dirichlet series are continued analytically over the whole plane and satisfy a certain additional assumption, then the error terms coming from the summatory functions of Dirichlet coefficients are good oscillation.

Keywords: arithmetical function, Dirichlet series, analytic continuation

Classification (MSC2000): 11M06

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Electronic fulltext finalized on: 8 Nov 2012. This page was last modified: 19 Nov 2012.

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© 2012 FIZ Karlsruhe / Zentralblatt MATH for the EMIS Electronic Edition