An explicit formula for sums of Ramanujan type sums
type: Item
instance of: scholarly article
title: An explicit formula for sums of Ramanujan type sums
zbMATH Open document ID: 0624.10006
author: Kenneth R. Johnson
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: For multiplicative arithmetical functions f and g \textit{D. R. Anderson} and \textit{T. M. Apostol} [Duke Math. J. 20, 211-216 (1953; Zbl 0050.042)] introduced the sum \(S_ k(n)=\sum_{d | (n,k)}f(d)\mu (k/d)g(k/d),\) which is multiplicative as a function of the subscripted variable k but (in general) nonmultiplicative in n. A well-known example of such a sum is Ramanujan's sum \(c_ k(n)=\sum_{d | (n,k)}d \mu (k/d).\) In the present paper the sums \(\sum_{d | n}S_ k(d)\) resp. \(\sum_{d | n}| S_ k(d)|\) are evaluated for completely multiplicative functions f, thus generalizing the author's evaluation of \(\sum_{d | n}| c_ k(d)|\) published earlier [Elem. Math. 38, 122-124 (1983; Zbl 0516.10002)]. The method is the same one as employed in the note mentioned above. The article contains an unusually large number of typographical errors.
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 4012708
zbMATH Keywords: generalizations of Ramanujan sum
zbMATH Keywords: completely multiplicative functions
zbMATH Keywords: evaluation
reviewed by: Joachim Herzog
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4012708