What, if anything, can be said about the distribution of $$ X_n = \sum_{k=1}^n \{\alpha\log k\} $$ where $\alpha$ is a nonzero real constant and $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$? In particular, are their constants $c_1,c_2,e_1,e_2,e_3$ such that $$ \limsup \frac{|X_n - c_1n|}{c_2n^{e_1}(\log^{e_2}n)^{e_3}} $$ is positive and finite? (And are they absolute, as I would guess, or do they vary based on $\alpha$?)
Possibly related to: Expected value of fractional part is less than 1/2, though I'm not making that claim of $c_1$.
Is this known to be an open problem? Should be be asking over on MathOverflow?