Continuity module of the distribution of additive functions related to the largest prime factors of integers (Q752071)

For \(\alpha >0\) let \(f(n)=\sum_{p\mid n}(\log p)^{\alpha}\). The distribution of \(f\) is studied with two renormalizations, one being \(v_x(n)=f(n)(\log x)^{-\alpha}\), the other \(T(n)=f(n)(\log P(n))^{- \alpha},\) where \(P(n)\) is the largest prime divisor of \(n\). Let \(F\) and \(G\) be the limiting distributions of \(v_x\) and \(T\), resp. Write \(Q_F(h)=\max F(y+h)-F(y),\) the modulus of continuity of \(G\); \(Q_G\) is defined similarly. Typical results are \[ Q_ F(h)\ll h^{1-1/\alpha}(\log 1/h)^{\alpha +1}\text{ for } \alpha >1, \] \[ Q_ G(h)\ll h^{1- 1/\alpha}(\log 1/h)^{\alpha +1}(\log \log 1/h)\text{ for } 1<\alpha \leq 2, \] \[ h^{1/\alpha}\ll Q_ F(h)\ll h^{1/\alpha}(\log 1/h)^{\alpha +1}\text{ for } \alpha >2. \]