The distribution law of divisors on a sequence of integers
Let \(f\), \(g\) be two nonnegative real-valued multiplicative functions such that \(|g(p^k)|\leq 1\) and \(|f(p^k)|\leq c\) for some constant \(c\) and for all \(k\in\mathbb{N}\). In addition, let these functions satisfy the following two analytic requirements related with parameters \(0<\alpha<\delta\leq 1\). {\parindent=0.7cm \begin{itemize}\item[{\(\bullet\)}]The series \(\sum_{p}\big(g(p)-\delta\big)/p^s, \text{Re}\,s>1\) has an analytic holomorphic continuation \(P_2(s)\) into the region \(\text{Re}\,s\geq 1-c_1/\log(|\text{Im}\, s|+3), \text{Im}\, s\in\mathbb{R}\), such that \(|P_1(s)|\leq(1-\delta_1)\log(|\text{Im}\, s|+2)+c_2\), where \(c_1>0\), \(c_2>0\) and \(\delta_1\in(0,1)\). \item[{\(\bullet\)}] The series \(\sum_{p}\big(g(p)/(1+f(p))-\alpha\big)/p^s, \text{Re}\,s>1\) has an analytic holomorphic continuation \(P_2(s)\) into the region \(\text{Re}\, s\geq 1-c_3/\log(|\text{Im}\, s|+3)\), \(\text{Im}\, s\in\mathbb{R}\), such that \(|P_2(s)|\leq(1-\delta_2)\log(|\text{Im}\, s|+2)+c_4\), where \(c_3>0\), \(c_4>0\) and \(\delta_2\in(0,1)\). \end{itemize}} The authors of the paper prove that for the described functions \(f, g\) \[ \frac{1}{G(x)}\sum\limits_{n\leq x}g(n)\frac{F(n,n^u)}{F(n)}=B(u,\delta-\alpha,\alpha)+O\Big((\log x)^{-\min\{\alpha,\delta-\alpha\}}\Big), \] uniformly with respect to \(x>1\) and \(0\leq u\leq 1\), where \(B\) denotes the beta distribution function, and \[ G(x)=\sum\limits_{n\leq x}g(n),\;F(n,n^u)=\sum\limits_{d\mid n,\,d\leq n^u}f(d),\;F(n)=\sum\limits_{d\mid n,\,d\leq n}f(d). \]
scientific article; zbMATH DE number 6529478