A note on some discrete valuation rings of arithmetical functions. (English) Zbl 1058.11007

Schwab, Emil Daniel; Silberberg, Gheorghe

Arch. Math., Brno 36, No. 2, 103-109 (2000).

Let \((A,+,*_D)\) be the ring of all arithmetical functions, where \(*_D\) is the Dirichlet convolution defined by \((f*_Dg)(n)=\sum _{d| {n}}f(d)g(\frac {n}{d}).\) The authors investigate properties of this ring. Namely they prove that this ring is neither Noetherian nor Artinian and it is not a valuation ring and they construct a non-Archimedean valuation with the discrete valuation ring \(B_D\) such that \(A\) is canonically embedded in \(B_D\).

Reviewer: Jiří Močkoř (Ostrava)

MSC:

11A25

Arithmetic functions; related numbers; inversion formulas

Keywords:

discrete valuation ring; arithmetical function; Dirichlet convolution