[Paper Review] Multiplicative function instead of logarithm (an elementary approach)

É. Yu. Lerner|ArXiv.org|Oct 10, 2007

Mathematical and Theoretical Analysis3 references4 citations

TL;DR

This paper establishes necessary and sufficient conditions under which a sequence of values of a multiplicative function over a finite field is most or almost most complicated under translation-invariant linear operators, generalizing Arnold's hypothesis that the logarithmic sequence is highly complex. Using elementary algebraic techniques, it proves that such sequences are maximally complex when the operator is divisible by the finite difference operator Δ, with key results on polynomial and multiplicative function dynamics in finite fields.

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.

Motivation & Objective

• To resolve V.I. Arnold's hypothesis that the logarithmic sequence is most complicated under the finite difference operator Δ in finite fields.
• To extend the notion of complexity beyond logarithmic sequences to include multiplicative functions.
• To identify necessary and sufficient conditions under which a multiplicative function generates a sequence of maximal or near-maximal complexity.
• To generalize Arnold's results on complexity for the operator Δ to a broader class of translation-invariant linear operators.
• To provide an elementary algebraic proof of the maximality of complexity for multiplicative functions, avoiding advanced tools.

Proposed method

• Define translation-invariant linear operators A on F_q^n that commute with the cyclic shift δ, ensuring invariance under cyclic permutations.
• Use the finite difference operator Δ = δ - I as a central object, and consider operators A divisible by Δ to generalize results.
• Analyze the dynamics of sequences generated by f(i) = multiplicative function values, focusing on preperiod and period length in the orbit of A.
• Apply polynomial modular arithmetic modulo y^n - 1 to study the algebraic structure of the generating functions F(y) = ∑f(i)y^i.
• Use the operator ⋅_k to analyze symmetry under multiplication by k coprime to n, showing invariance of coefficient distribution in H(y) = F(y)F^−(y).
• Prove that ∑h(i) = 0 and h(1) = h(2) = ... = h(n-1) by exploiting multiplicative function properties and group-theoretic structure modulo y^n - 1.

Experimental results

Research questions

• RQ1 Under what conditions is a sequence generated by a multiplicative function in a finite field maximally complex under a translation-invariant operator?
• RQ2 Is Arnold’s hypothesis that the logarithmic sequence is most complicated for Δ valid for a broader class of operators?
• RQ3 Can the complexity of multiplicative functions be characterized algebraically via their generating polynomials modulo y^n - 1?
• RQ4 What is the role of the operator Δ in determining the maximal preperiod and period length of sequences?
• RQ5 Why does the failure of invertibility in the algebra modulo ∑y^i imply that GCD(F(y), ∑y^i) = 1, and how does this relate to maximality of complexity?

Key findings

• A multiplicative function f: F_q → F_q generates a sequence of maximal complexity (most complicated) if and only if the operator A is divisible by Δ and n is coprime to the characteristic p of the field.
• The sequence f(i) = Legendre symbol (i/r) mod r (quadratic residue indicator) is most complicated for Δ when r = n+1 is an odd prime, confirming Arnold’s hypothesis.
• For q=2 and n odd, the graph of Δ: F_2^n → F_2^n decomposes into cycles O_m and trees T_2, with the number of components growing rapidly with n, e.g., 34636834 components for n=31.
• The coefficient sequence h(i) of H(y) = F(y)F^−(y) mod (y^n - 1) satisfies h(1) = h(2) = ... = h(n-1), and h(0) - h(1) ≠ 0, proving invertibility modulo ∑y^i.
• The GCD condition ∑_{i=1}^{n-1}f(i)y^i and ∑_{i=0}^{n-1}y^i is 1, which is equivalent to the sequence being most complicated, and this holds precisely when f is a nontrivial multiplicative function with f(-1) ≠ 0.
• The result fails if A is not divisible by Δ; for example, A = I + δ + ... + δ^{n-1} yields shorter periods for nontrivial multiplicative functions, invalidating the maximality claim.