Mean values of multiplicative functions with weight. (Q1810002)

Let \(f\) denote a complex-valued multiplicative arithmetic function. Various authors have established necessary and sufficient conditions under which the mean value \(M(f)\) of \(f\) exists and is non-zero. Two directions in which this study was pursued are particularly relevant to this paper. In [Acta Arith. 42, 21--47 (1982; Zbl 0493.10047)], \textit{B. V. Levin} and \textit{N. M. Timofeev} generalized the method by G. Halász in 1968, in which the Dirichlet series with coefficients \(f(n)\), \(| f(n)|\leq 1\), was compared with \(\zeta(s)\), by replacing \(\zeta(s)\) by another Dirichlet series with nonnegative multiplicative coefficients. On the other hand, the necessary and sufficient conditions obtained by \textit{K.-H. Indlekofer} in [Math. Z. 172, 255--271 (1980; Zbl 0416.10035)] concerned the convergence of several infinite series with summands involving \(f(p)\) or \(f(p^r)\) for \(p\) prime. The main result in the current paper combines these two approaches by introducing a nonnegative multiplicative function \(\rho\) in a certain class and investigating \(M_\rho(f)= \lim_{x\to\infty} M_\rho(f, x)\) where \[ M_\rho(f, x)= (S(\rho, x))^{-1} \sum_{n\leq x} f(n)\rho(n) \] with \(S(\rho,x)= \sum_{n\leq x}\rho(n)\). Here the function \(f\) belongs to a wider class than that considered by Indlekofer whose result follows from the case \(\rho(n)= 1\). The theorem is too complicated to quote here but it concerns functions \(f\) satisfying \[ \Biggl(\limsup_{x\to\infty}\, (S(\rho, x))^{-1} \sum_{n\leq x}| f(n)|^\alpha \rho(n)\Biggr)^{1/\alpha}< \infty, \] where \(\alpha\geq 1\) and another condition. A crucial tool in the proof is the establishment of necessary and sufficient conditions for a real additive function \(g(n)\) to be finitely distributed with weight \(\rho(n)\) for \(\rho\) belonging to a certain class of nonnegative multiplicative functions, this being the analogue of the Erdős criterion for the finite distribution of additive functions.