Geodesic
On a Relation Between Sums of Arithmetical Functions and Dirichlet Series
Publications de l'Institut Mathématique, _N_S_92 (2012) no. 106, p. 97

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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.
Classification : 11M06
Keywords: arithmetical function, Dirichlet series, analytic continuation 
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     title = {On a {Relation} {Between} {Sums} of {Arithmetical} {Functions} and {Dirichlet} {Series}},
     journal = {Publications de l'Institut Math\'ematique},
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Hideaki Ishikawa; Yuichi Kamiya. On a Relation Between Sums of Arithmetical Functions and Dirichlet Series. Publications de l'Institut Mathématique, _N_S_92 (2012) no. 106, p. 97 . http://geodesic.mathdoc.fr/item/PIM_2012_N_S_92_106_a6/