Some notes on the pseudorandomness of Legendre symbol and Liouville function

TL;DR

This paper investigates the pseudorandom properties of the Legendre symbol and Liouville function, providing improved bounds and extending results to polynomial and binary arithmetic functions.

Contribution

It advances the understanding of pseudorandomness by improving bounds on Boolean representations and linear complexity for these functions, including generalizations to polynomial and binary cases.

Findings

Improved bounds on the degree and sparsity of Boolean functions representing the Legendre symbol.

Enhanced bounds on the Nth linear complexity of the Legendre sequence.

Results extended to Liouville functions over integers and polynomials, and to general binary arithmetic functions.

Abstract

We improve bounds on the degree and sparsity of Boolean functions representing the Legendre symbol as well as on the $N$th linear complexity of the Legendre sequence. We also prove similar results for both the Liouville function for integers and its analog for polynomials over $\mathbb{F}_2$, or more general for any (binary) arithmetic function which satisfies $f(2n)=-f(n)$ for $n=1,2,\ldots$