Multiplicative function instead of logarithm (an elementary approach)

TL;DR

This paper explores the complexity of sequences generated by multiplicative functions over finite fields, extending Arnold's hypothesis about logarithmic sequences, and establishes conditions for their maximal complexity in a broader class of operators.

Contribution

It provides necessary and sufficient conditions for multiplicative function sequences to be most complicated under certain operators, broadening Arnold's original hypothesis.

Findings

Sequences of multiplicative functions can be most complicated for operators divisible by differentiation.

The paper generalizes Arnold's hypothesis to a wider class of operators.

Conditions for maximal complexity are explicitly characterized.

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.