Let $d_k(n) = \sum_{ n_1 \cdots n_k = n}1$ be the $k$-fold divisor function. We call a function $f:\mathbb{ N} \to \mathbb{ C}$ a $d_k$-bounded multiplicative function, if $f$ is multiplicative and $|f(n)| \leq d_k(n)$ for every $n \in \mathbb{ N}$. In this paper we improve Mangerel's results which extend the Matom\"aki-Radziwi{ \l\l} theorem to divisor bounded multiplicative functions. In particular, we prove that for sufficiently large $X \geq 2$, any $\epsilon>0$ and $h \geq (\log X)^{ k \log k - k + 1 + \epsilon}$ , we have $$ \frac{ 1}{ h}\sum_{ x<n\leq x+h}d_k(n)-\frac{ 1}{ x}\sum_{ x<n\leq 2x}d_{ k}(n) = o(\log^{ k-1} x) $$ for almost all $x \in [X,2X]$. We also demonstrate that the exponent $k \log k-k+1$ is optimal.
Categories: math.NT