Sándor, J.; Sivaramakrishnan, R. The many facets of Euler’s totient. III: An assortment of miscellaneous topics. (English) Zbl 0785.11005 Nieuw Arch. Wiskd., IV. Ser. 11, No. 2, 97-130 (1993). As the title suggests, some miscellaneous properties of the Euler totient function \(\varphi\) are studied. Certain remarks on inequalities connected with \(\varphi\) are stated and some new inequalities are pointed out. The core-reduced totient is introduced and is shown to be yet another multiplicative function connected with the ring \(\mathbb{Z}_ r\) of integers \(\pmod r\). A new relation between \(\varphi\) and the Dedekind totient \(\psi\) is given in terms of an arithmetical function \(\overline{\Lambda}\), which is shown to vanish at prime powers. This means that \(\overline{\Lambda}\) behaves like an ‘anti-Mangoldt’ function. Also some other identities and series expansions are given.For parts I and II in this series of papers about Euler’s totient, see the second author [ibid. 4, 175-190 (1986; Zbl 0634.10006)] and [ibid. 8, 169-187 (1990; Zbl 0721.11002)]. Reviewer: P.Haukkanen (Tampere) MSC: 11A25 Arithmetic functions; related numbers; inversion formulas Keywords:Euler totient function; inequalities; core-reduced totient; multiplicative function; Dedekind totient; identities; series expansions Biographic References: Fenchel, Werner; Freudenthal, Hans Citations:Zbl 0634.10006; Zbl 0721.11002 PDFBibTeX XMLCite \textit{J. Sándor} and \textit{R. Sivaramakrishnan}, Nieuw Arch. Wiskd., IV. Ser. 11, No. 2, 97--130 (1993; Zbl 0785.11005)