International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 351-354
doi:10.1155/S0161171288000419
An identity for a class of arithmetical functions of two variables
Department of Mathematics, The University of Toledo, Toledo 43606, Ohio , USA
Received 17 November 1986
Copyright © 1988 J. Chidambaraswamy and P. V. Krishnaiah. 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.
Abstract
For a positive integer r, let r∗ denote the quotient of r by its largest squarefree divisor (1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1) or 0according as r∗|n or not where C(n,r) is the well known Ramanujan's sum. In this paper, using a different method, we generalize (∗) to a wide class of arithmetical functions of 2 variables and deduce as special cases (∗) and similar formulae for several generalizations of Ramanujan''s sum.