International Journal of Mathematics and Mathematical Sciences
Volume 19 (1996), Issue 2, Pages 209-217

Some characterizations of totients

Pentti Haukkanen

Department of Mathematical Sciences, University of Tampere, P.O. Box 607, Tampere FIN-33101, Finland

Received 4 October 1994; Revised 27 January 1995

Copyright © 1996 Pentti Haukkanen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.