Multiplicative function instead of logarithm (an elementary approach)

Abstract

V.I. Arnold has recently defined the complexity of finite sequences of zeroes and ones in terms of periods and preperiods of attractors of a dynamic system of the operator of finite differentiation. Arnold has set up a hypothesis that the sequence of the values of the logarithm is most complicated or almost most complicated. In this paper we obtain the necessary and sufficient conditions which make this sequence (supplemented with zero) most complicated for a more wide class of operators. We prove that a sequence of values of a multiplicative function in a finite field is most complicated or almost most complicated for any operator divisible by the differentiation operator.