Tauberian theorems for sequences linked by a convolution. II (Q1961702)

[Part I, cf. Math. Nachr. 193, 211-234 (1998; Zbl 0936.11051).] The author deals with arithmetical functions \( g, \lambda: {\mathbb N} \cup \{0\} \to {\mathbb R}, \) satisfying \( g(0) = 1, \lambda \geq 0, \) and \[ (g * \lambda)(n) \left[ :=\sum_{\nu = 0}^n g(\nu) \cdot \lambda (n-\nu)\right] = g^\prime(n) : = (n+1) g(n+1). \] Assuming some information concerning \( G(n) :=\sum_{\nu=0}^n g(\nu) \) for \( n \to \infty \), the author obtains results on the summatory function \( L(n) := \sum_{\nu =0}^n \lambda(\nu). \) By Abelian and Tauberian arguments, the author shows: 1) If \( 0 < \liminf_{n \to \infty} {G(n) \over n+1} \leq \limsup_{n \to \infty} {G(n) \over n+1} < \infty, \) then (with explicitly given constants \( a, b \)) \[ \liminf_{n \to \infty} {L(n) \over n+1} \geq 2(b-\sqrt{b(b-a)}), \quad \limsup_{n \to \infty} {L(n) \over n+1} \leq 2(b+\sqrt{b(b-a)}). \] 2) For \( 0 < A < \infty \) \[ \lim_{n \to \infty} {G(n) \over n+1} = A \quad\text{ if and only if }\quad\sum^\infty_{m=1} {\lambda(m-1)-1 \over m} = \log A. \] 3) If \[ {G(n) \over n+1} = A + O(n^{-\alpha}), \quad\text{ where } 0 < A < \infty,\quad 0 < \alpha \leq 1, \] then \[ {L(n) \over n+1} = 1 +O \left( n^{-{1 \over 2}\alpha} \right) . \]