On divisor bounded multiplicative functions in short intervals

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 $\mathrm{\mid }f(n)\mathrm{\mid }\le d_k(n)$ for every $n\in \mathbb{N}$. In this paper we improve Mangerel’s results which extend the Matomäki-Radziwi{łł} theorem to divisor bounded multiplicative functions. In particular, we prove that for sufficiently large $X\ge 2$, any $\epsilon >0$ and $h\ge (\log X{)}^{k\log k-k+1+\epsilon }$ , we have $$\frac{1}{h}\sum _{x<n\le x+h}{d}_k(n)-\frac{1}{x}\sum _{x<n\le 2x}{d}_k(n)=o({\log }^{k-1}x)$$ for almost all $x\in [X,2X]$. We also demonstrate that the exponent $k\log k-k+1$ is optimal.