ABSTRACT
It was E. T. Bell (1883–1960) who considered the multiplicative inverse of an arithmetic function. In his paper entitled: An arithmetical theory of certain numerical functions (University of Washington Publications in Mathematical and Physical Sciences Vol No.1 (1915)), Bell shows that an arithmetic function f possesses a Dirichlet inverse if, and only if, f (1) 6= 0. In October 1927, R. Vaidyanathaswamy showed the existence of the inverse of a multiplicative function independently. In the Journal of Indian Math. Society (Notes and questions) 17 (1927), 69–73, Vaidynathaswamy established that every multiplicative function of one variable possesses an inverse which is also multiplicative. Only later, when he was in St. Andrews University, (while he was with Professor H. W. Turnbull), he got interested in the papers of E. T. Bell. Vaidynathaswamy refers to the work of Bell in his memoir: ‘The theory of multiplicative arithmetic functions’ (Trans. Amer. Math. Soc. 33 (1931) 579–662). It is to be remarked that Bell recognized the algebraic foundations of the theory of arithmetical functions and made use of Cauchy composition given by h(n,r) =
f (a)g(b) and other techniques. See Bell: Euler Algebra (Trans. Amer. Math. Soc. 25 (1923) 135–154) and Modular interpolation (Bull. Amer. Math. Soc., 37 (1931) 65–68). Since then, the algebraic approach to the theory of arithmetic functions came to be known better resulting in further work by L. Carlitz (1907–1999), Eckford Cohen, P. Kesava Menon and others.
