An explicit formula for sums of Ramanujan type sums
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.
scientific article; zbMATH DE number 4012708