On irregularities of distribution of weighted sums-of-digits

Let s(n)s(n) be the sum-of-digits function of nn in base 22. Newman (1969) [9] has shown the surprising phenomenon that #{0≤nc⋅Nlog3log4 for all NN (with some c>0c>0).Motivated by investigations of distribution properties of Niederreiter–Halton sequences we study #{0≤n