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On the concentration of certain additive functions

Dimitris Koukoulopoulos (2014)

Acta Arithmetica

We study the concentration of the distribution of an additive function f when the sequence of prime values of f decays fast and has good spacing properties. In particular, we prove a conjecture by Erdős and Kátai on the concentration of f ( n ) = p | n ( l o g p ) - c when c > 1.

On the counting function for the generalized Niven numbers

Ryan Daileda, Jessica Jou, Robert Lemke-Oliver, Elizabeth Rossolimo, Enrique Treviño (2009)

Journal de Théorie des Nombres de Bordeaux

Given an integer base q 2 and a completely q -additive arithmetic function f taking integer values, we deduce an asymptotic expression for the counting function N f ( x ) = # 0 n < x | f ( n ) n under a mild restriction on the values of f . When f = s q , the base q sum of digits function, the integers counted by N f are the so-called base q Niven numbers, and our result provides a generalization of the asymptotic known in that case.

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