Sums of some multiplicative functions over a special set of integers (Q2773341)

The authors prove several general and important results on sums of the form NEWLINE\[NEWLINE F_g(x) = \sum_{g(n)\leq x}f(n), NEWLINE\]NEWLINE where \(f(n), g(n)\) are two multiplicative functions such that \(f\) is ``small'' (embodied in the condition \(|f(p) - \kappa|\leq C_1p^{-\eta}\) for all primes \(p\), \(|\kappa|< 1/\eta\), \(\eta > 0)\) and \(g(n)\) is ``large'' (roughly \(g(p)\) is of the order \(\alpha p^\theta (\alpha, \theta > 0)\)). This work builds on results and methods of several previous authors, most notably on the paper of \textit{M. Balazard} and \textit{G. Tenenbaum} [Compos. Math. 110, 239--250 (1998; Zbl 0893.11037)]. The main result is Theorem 1, which (under precise conditions stated in the text) says that, for any integer \(J\geq 0\), NEWLINE\[NEWLINE F_g(x) = {x^{1/\theta}\over(\log x)^{1-\kappa/\alpha^{1/\theta}}} \left\{\sum_{j=0}^J{P_j(\log\log x)\over\log^jx} + O(R_{J,\lambda}(x))\right\}, NEWLINE\]NEWLINE where the coefficients of the polynomial \(P_j\) (of degree \(j\)) are explicitly evaluated, and the error term \(R_{J,\lambda}(x)\) satisfies an explicit, sharp bound. The ``sharpness'' is manifested in the fact that, among other results, Theorem 1 yields the strongest known form of the prime number theorem. It also yields results on local densities (Theorems 2 and 3), namely the asymptotic evaluation of the sums NEWLINE\[NEWLINE\sum_{g(n)\leq x,\;\Omega(n)=k}f(n),\qquad\sum_{g(n)\leq x,\;\omega(n)=k}f(n), NEWLINE\]NEWLINE where \(\Omega(n)\) is the total number of prime factors of \(n\), while \(\omega(n)\) is the number of distinct prime factors of \(n\). This is done by using Theorem 1 with \(f(n)\) replaced by \(f(n)z^{\Omega(n)}\) or \(f(n)z^{\omega(n)}\) with the complex variable \(z\) lying in a suitable circle with center at the origin. The proofs depend on analytic properties of the appropriate generating Dirichlet series. This is furnished by Theorem 4.