Exponential divisor functions

Abstract

Consider the operator E on arithmetic functions such that Ef is the multiplicative arithmetic function defined by (Ef )(p a ) = f (a) for every prime power p a . We investigate the behaviour of E m τ k , where τ k is a kdimensional divisor function and E m stands for the m-fold iterate of E. We estimate the error terms of n x E m τ k (n) for various combinations of m and k. We also study properties of E m f for arbitrary f and sufficiently large m.Our study provides a unified approach to functions with exponential divisors. We improve special cases of the Dirichlet asymmetric divisor problem and several results on the exponential divisor and totient functions.