Asymptotic expansions of some arithmetic sums (Q1311076)

Given a nonnegative multiplicative function \(\beta: \mathbb{N}\to [0,\infty[\), and a nonnegative additive function \(\alpha\), the author is interested in connections between asymptotic expansions of the sums \[ \mathbb{N}_{\alpha,\beta}(x)= \sum_{n; \alpha(n)\leq x} \beta(n) \qquad \text{and} \qquad P_{\alpha,\beta}(x)= \sum_{p^ m; \alpha(p^ m)\leq x} \beta(p^ m). \] The author uses the following notation: \(Q\) is the class of functions having an asymptotic ``\(Q\)-expansion'' \[ f(x)\sim \sum_{n=1}^ \infty e^{s_ n x} x^{\tau_ n-1} Q_ n(\log x), \qquad x\to\infty, \] where \(s_ n,\tau_ n\in\mathbb{C}\), \(\text{Re } s_ n=1\), \(\text{Re } \tau_ n\to-\infty\), all of the pairs \((s_ n, \tau_ n)\) are distinct and \(Q_ n(u)\) are polynomials in \(\mathbb{C}[u]\). \(Q_ 1\subset Q\) is the subclass of functions with a ``\(Q\)-expansion'' of the form \[ f(x)\sim \sum_{\tau_ n=0} q_ n e^{s_ n x} x^{- 1}+ \sum_{\text{Re } \tau_ n<0} e^{s_ n x} x^{\tau_ n-1} Q_ n(\log x). \] Denoting by \(\beta^*\) a complex-valued function satisfying \(|\beta^*|\leq \beta\), and assuming in addition, that \(\sum_ p (\sigma_ p^{(k)}(1))^ 2<\infty\) for any \(k=1,2,\dots,\) where \(\sigma_ p(s)= \sum_{m=1}^ \infty \beta(p^ m) e^{-s\alpha(p^ m)}\), the author's main result is the following: \[ \begin{alignedat}{2} \text{If}\quad &P_{\alpha,\beta}\in Q_ 1, \quad \text{then} &&\quad \mathbb{N}_{\alpha,\beta}\in Q; \tag{1}\\ \text{If} \quad &P_{\alpha,\beta^*}\in Q_ 1, \quad \text{and} &&\quad \mathbb{N}_{\alpha,\beta} \in Q, \quad \text{then} \quad \mathbb{N}_{\alpha,\beta^*}\in Q.\tag{2} \end{alignedat} \] Moreover, sufficient conditions are given that certain \(Q\)-expansions for \(P_{\alpha,\beta}\) and \(\mathbb{N}_{\alpha,\beta}\) are equivalent. In particular this result extends a theorem of \textit{R. Balasubramanian} and \textit{K. Ramachandra} [Acta Arith. 49, 313-322 (1988; Zbl 0633.10039)].