Tauberian theorems for sequences satisfying certain convolution relations (Q2770426)

Let \(\mathcal{F}\) be the complex algebra of arithmetic functions \(f:{\mathbb{N}}\to {\mathbb{C}}\) endowed with usual linear operations and the Cauchy convolution \(*\), and \(\mathcal{R}\) the subalgebra of \(\mathcal{F}\) of all homogeneous recurrent sequences. Given a \(g\in\mathcal{F}\) let \(g^\prime\) be defined by \(g^\prime(n)=(n+1)f(n+1)\) for \(n\in{\mathbb{N}}\), and \(\lambda\) by \(g^\prime=\lambda*g\). Motivated by results proved by \textit{R. Warlimont} [Math. Nachr. 193, 211-234 (1998; Zbl 0936.11051)] the author proves some Tauberian theorems for \(\lambda\) without presuming some assumptions on \(\lambda\). The first idea behind the proofs is to approximate the \(g\) by a recurrent sequence \(\widetilde{g}\in\mathcal{R}\). For instance, if (1) \(\sum_{n=0}^\infty |g(n)-\widetilde{g}(n)|<\infty\), (2) \(g(n)=\widetilde{g}(n)+o(n^{-1})\), (3) the generating power series \(G(z)\) of \(g\) converges and is zero-free in the closure \(\bar{U}\) of the open unit disc \(U\) and (4) the roots \(\{\xi_j\}_{j=1}^k\) of the companion polynomial of \(\widetilde{g}\) lie in \(\partial U\), then \(\lambda(n)=\sum_{j=1}^k \nu_j\xi_j^{n+1}+o(1)\), \(n\to\infty\), where \(\nu_j\) stands for the multiplicity of \(\xi_j\). The author shows that condition (1) is not sufficient for boundedness of \(\lambda\). However, (1), (3) and (4) yield that \(t(n)=\lambda(n)-\sum_{j=1}^k \nu_j\xi_j^{n+1}\) satisfies \(\lim_{n\to\infty} (1/n)\sum_{\nu=0}^n |t(\nu)|=0\), to mention explicitly some of the author's results. Using \textit{L. Lucht} and \textit{K. Reifenrath}'s [Analytic Number Theory, Proc. Conf. in Honor of H. Halberstam, Birkhäuser, Boston, Vol. 2, Prog. Math. 139, 607-619 (1996; Zbl 0853.11066)] generalization of Wiener's inversion theorem, the author proves also (optimal) quantitative Tauberian theorems.