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 $n=1,2,\ldots$

翻译：我们改进了表示勒让德符号的布尔函数的次数与稀疏性上界，以及勒让德序列的第$N$次线性复杂度上界。我们还对整数刘维尔函数及其在$\mathbb{F}_2$上多项式的模拟（或更一般地，对任何满足$f(2n)=-f(n)$（$n=1,2,\ldots$）的（二元）算术函数）证明了类似的结果。