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On a relation between sums of arithmetical functions and Dirichlet series. (English) Zbl 1274.11144

The authors introduce a concept called good oscillation. A function is called good oscillation, if its \(m\)-tuple integrals are bounded by functions having mild orders. They 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. The authors 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. As an application they consider the multiple Dirichlet series generated by characters \(\chi_k(n)\) (\(k =1,\dots,j\)) to the same modulus \(q\) (\(\geq 2\)).

MSC:

11N37 Asymptotic results on arithmetic functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
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