Inequalities for divisor functions (Q543564)

The author proves inequalities of the divisor function \(\tau(n)= \sum_{d|n}1\), which are of the form \[ \tau(n)\underset{\delta,\beta}{}\ll \sum_{d|n,d\leq n^\delta}\tau(d)^\beta \] for square-free \(n\); here \(0< \delta\leq{1\over 2}\) and \[ \beta> {\log\delta\over\log 2}+{1\over\delta} \Biggl(1+(1-\delta){\log(1- \delta)\over\log 2}\Biggr). \] This improves results of Landau, Friedlander and Iwaniec \((\delta={1\over k})\).