Modeling the beta distribution in short intervals (Q2313192)

Let \(f,g:\mathbb{N}\rightarrow\mathbb{C}\) be two multiplicative functions. Denote: \[ T_f(m,v)=\sum_{d|m,d\leqslant v}f(d),\ m\in\mathbb{N}, v\in\mathbb{R},\ T_f(m)=T_f(m,m), \] \[G(x,y,g)=\sum_{x<n\leqslant x+y}g(n),\ x,y\geqslant 0, \] \[ F(x,y,u,g,f)=\frac{1}{G(x,y,g)}\sum_{x<m\leqslant x+y}\frac{T_f(m,m^u)}{T_f(m)}g(m). \] The authors of the paper obtain upper bound for distance between \(F(x,y,u,f,g)\) and a suitable \textit{Beta distribution} for a wide class of multiplicative functions \(f,g\). The derived upper bound is uniform in \(u\in[0,1]\). Parameters of the limiting \textit{Beta distribution} depend on the behavior of functions \(f,g\) on prime numbers.