A Turán-Kubilius inequality with multiplicative weights
The authors prove a weighted version of the Turán-Kubilius inequality: Assume that \(g\) is a complex-valued additive arithmetical function. Then \[ \sum_{n\leq x}f(n)\cdot|g(n)-A_f(x)|^2\;\ll\;M(x)\cdot D^2_f(x),\tag \(*\) \] where \(f\) is a nonnegative multiplicative arithmetical weight function, satisfying \[ \sum_p\;\sum_{r\geq 2}{{f(p^r)}\over{p^r}} \ll\;1,\qquad \sum_{p^r\leq y}f(p^r)\;\ll\;{y\over{\log y}},\tag \(**\) \] and there exist constants \(C_1,C_2\), such that for all \(y\geq C_1\), \(y'\geq y+C_2\cdot{y\over {\log\log y}}\), \[ \sum_{y\leq p\leq y'}f(p)\;\gg\;{{y'-y}\over{\log y'}}.\tag \(***\) \] The implied constant in \((*)\) depends only on those constants occurring in \((**)\) and \((***)\). The usual notations \(A(x)\), \(D(x)\) are changed to \[ A_f(x)={1\over{M(x)}}\cdot \sum_{n\leq x}f(n)g(n)= {1\over{M(x)}}\cdot \sum_{p^r\leq x}f(p^r)g(p^r)\cdot M_p\biggl({x\over p^r}\biggr), \] and \[ D^2_f(x)= \sum_{p^r\leq x}|g^2(p^r)|\cdot f(p^r)\cdot {{M(x/p^r)}\over {M(x)}}, \] where \(M(x)=\sum_{n\leq x}f(n)\), \(M_k(x)= \sum_{\substack{ n\leq x\\ (n,k)=1}} f(n)\). As an application the authors prove a generalized version of Delange's theorem: Suppose that the weight function \(f\) satisfies the conditions of the main theorem, and that \(\sum_p{{f(p)-\Phi(p)}\over p}\) converges, where \(\Phi\) is a complex-valued multiplicative function, \(|\Phi|\leq f\). Then \[ \lim_{x\to\infty} {1\over{M(x)}}\cdot\sum_{n\leq x}\Phi(n)=C \] exists, where \(C\) is given by a product representation. Finally, the authors prove a generalized Erdös-Wintner theorem with weights for real-valued additive functions, giving necessary and sufficient conditions (convergence of three series) for the convergence of \[ {1\over{M(x)}}\cdot \sum_{\substack{ n\leq x\\ g(n)\leq t}} f(n), \qquad x\to\infty, \] to a distribution function.
multiplicative functions in short intervals
generalized Delange theorem for multiplicative functions
nonnegative multiplicative arithmetical weight function
generalized Erdös-Wintner theorem with weights for real-valued additive functions
scientific article; zbMATH DE number 878114