Erd(?) S-Kac-type Theorem With Weight D_k(n) On Short Interval And Mean Value Problem Related To D_k(f) On Function Domain

For integer k?2,let d _k(n) be the k-fold divisor function,which denotes the number of solutions ofn=n₁n₂…n_k,where n₁,n₂,…,n_k are oositive integers.By using the Selberg-Delange method and the Berry-Esseen inequality,we study the mean value of d_k(n)z^?(n) on short intervals,and obtain the asymototic formula.Using this asymototic formula,we futher orove an Erd(?)s-Kac tyoe theorem on short intervals weighted by d _k(n),in which the error term is ootimal.We also study the mean value oroblem related to thek-fold divisor function d_k(f) in function field F_q(T),which refers to the number of solutions off=f₁f₂…f_k,where f₁,f₂,…,f_k are monic oolynomials.By establishing a suitable Hankle contour and using the residue theorem,we give an asymototic formula for the mean value of d_k(f)z^?(f).Our results generalize a recent result of K.Liu and J.Wu.