On the upper bound of the $L_2$-discrepancy of Halton’s sequence

Abstract

Let $ (H(n))_{n \geq 0} $ be a $2$-dimensional Halton’s sequence. Let $D_{2} ( (H(n))_{n=0}^{N-1}) $ be the $L_2$-discrepancy of $ (H_n)_{n=0}^{N-1} $. It is known that $$\limsup _{N \to \infty } (\log N)^{-1} D_{2} ( H(n) )_{n=0}^{N-1} \gt 0.$$ In this p