Exponential divisor functions

Original · EN

Consider the operator E on arithmetic functions such that Ef is the multiplicative arithmetic function defined by (Ef)(pᵃ) = f(a) for every prime power pᵃ. We investigate the behaviour of Eᵐτₖ, where τₖ is a k-dimensional divisor function and Eᵐ stands for the m-fold iterate of E. We estimate the error terms of ∑ₙ≤ ₓ Eᵐτₖ(n) for various combinations of m and k. We also study properties of Eᵐf 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.