Some notes on the pseudorandomness of Legendre symbol and Liouville function

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 $n=1,2,\ldots$