Mean-value theorems for multiplicative functions on additive arithmetic semigroups via Halász's method
The author is interested in mean-value theorems for multiplicative functions \(f: {\mathcal G}\to {\mathbb C}\), defined on an additive arithmetical semigroup \(({\mathcal G},\partial)\) with an (additive) degree map \(\partial:{\mathcal G}\to {\mathbb N}\cup \{0\}\). \(G(n)\) denotes the number of elements \(a\) in \(\mathcal G\) of degree \(\partial(a)=n\). Primes in \(\mathcal G\) are denoted by the letter \(p\). The author assumes that, with real constants \(A_\rho\), \(A_r>0\), \(\rho_1<\rho_2<\dots<\rho_r\), the asymptotic formula \[ G(n) = q^n \cdot \sum_{\nu=1}^r A_\nu n^{\rho_\nu -1} + {\mathcal O}\left(q^n n^{-\gamma}\right) \] holds (for \(n\to\infty\)), where \(\gamma > 1+ \rho_r\). Using \textit{G. Halász}'s method [Acta Math. Acad. Sci. Hung. 19, 365--403 (1968; Zbl 0165.05804)], the author proves the following main result (Theorem 1): Assume that (with positive constants \(\delta\) and \(c_1\)) \[ \sum_p | f(p) |^{1+\delta}\partial(p) r^{\partial(p)} \leq {c_1 \over {1-qr}} \;\hbox{ for } 0\leq r < {1\over q}, \] that \[ \sum_p \sum_{k \geq 2} | f(p^k)| q^{-k\partial(p)} \, \hbox{ is convergent}, \] and that, for all primes \(p \in {\mathcal G}\) and all reals \(\theta\), \[ 1 + \sum_{k\geq 1} f(p^k) q^{-k\partial(p) \cdot (1+i\theta)} \not=0. \] Then (with some slowly oscillating function \(L\) satisfying \(| L| = 1\), and with positive constants \(\alpha,\tau\)) the asymptotic formula \[ \sum_{a\in {\mathcal G}} f(a) y^{\partial(a)} \left[ = \prod_p \left( 1+\sum_{k\geq 1} f(p^k) y^{k\partial(p)}\right)\; \right] = \qquad\qquad\qquad \] \[ = {c \over{(1-q^{1+i\alpha}y)^\tau}} \cdot L \left( {1\over {1-q| y|}} \right) + o\left( {1\over {(1-q| y|)^\tau}} \right) \] (for \(| y| \to q^{-1})\) implies that \[ \sum_{\partial(a) =m} f(a) = {{c q^{m(1+i\alpha)}\cdot m^{\tau-1}} \over {\Gamma(\tau)}} \cdot L(m) + o(q^m m^{\tau-1}), \] as \(m\to\infty\). In Theorem 2 it is proved that the mean-value is zero, if -- in addition to the first two conditions of Theorem 1 -- \[ \sum_{m\geq 1} q^{-m} \left( \sum_{p, \partial(p)=m} 1 - \Re\left( e^{im\theta} \sum_{\partial (p)=m} f(p) \right) \right) \] diverges to \(\infty\) uniformly for \(|\theta| \leq\pi\). Theorem 3 gives a semigroup-analogue of a theorem of E. Wirsing (1967). Finally, Theorem 4 proves the existence of a non--zero mean-value \[ m_f = \prod_p \bigl( 1-q^{-\partial(p)}\bigr) \cdot \left( 1+ \sum_{k\geq1} f(p^k) q^{k\partial(p)}\right) \] if the following conditions are satisfied: \[ \sum_p \sum_{k\geq 2} | f(p^k) | q^{- k \partial(p) }<\infty, \] \[ \sum_{p,\; | f(p) | > 1+\eta} | f(p)| q^{-\partial(p)} < \infty, \qquad \sum_{p,\; | f(p) | \leq 1+\eta} | f(p)-1|^2 q^{-\partial(p)} < \infty, \] [for some \(\eta > 0\)], \[ 1 +\sum_{k\geq 1} f(p^k) q^{-k\partial(p)} \not= 0 \quad\hbox{ for all } p, \] and \[ \sum_{m\geq 1} q^{-m} \left( \sum_{p, \partial(p)=m} 1 - \sum_{\partial(p)=m} f(p) \right) \] is convergent. Moreover, the author also gives converse results (if the mean-value exists, then \dots\ \dots).
multiplicative functions on additive arithmetic semigroups
scientific article; zbMATH DE number 1968533