Exponential divisor functions

Author:

Andrew~V.~Lelechenko

2010 Mathematics Subject Classification:

11N37, 11M06, 11A25.

Keywords:

average order, exponential divisor function, generalized divisor function.

Abstract:

We 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\tau_k, where \tau_k is a k-dimensional divisor function and E^m stands for the m-fold iterate of E. We estimate the error terms of \sum_{n\leqslant x} E^m\tau_k(n) for various combinations of m and k. We also study properties of E^mf for arbitrary f and sufficiently large m. We improve special cases of the Dirichlet asymmetric divisor problem and several results on the exponential divisor and totient functions, also.

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lel_2015

Vol. 10 (18), 2015