Tauberian theorem of Erdős revisited (Q5955199)

The aim of this paper is to present a simplified proof of the following Tauberian remainder theorem of \textit{P. Erdős} [J. Indian Math. Soc., n. Ser. 13, 131-144 (1949; Zbl 0034.31501)] saying that if \(a_n\geq 0\), \(n \geq 1\), then we have with \(s_n=\sum_{k=1}^n a_k\) \((n \in \mathbb{N})\) that \[ \sum_{k=1}^na_k(s_{n-k}+k)=n^2+O(n) \quad\text{implies}\quad s_n=n+O(1) \] (note the strong remainder term). See also the paper by \textit{H. N. Shapiro} [Commun. Pure Appl. Math. 12, 579-610 (1959; Zbl 0101.03203)]. The rather involved original proof uses combinatorial arguments, whereas the present proof uses a fundamental relation, attributed by the author to C. L. Siegel, which allows a well-structured simpler proof presented in this paper. In addition a supplementary Tauberian result is proved, where the remainders in the implication above are \(O(n^{1+\gamma})\) and \(O(n^\gamma)\) \((0<\gamma<1)\), respectively.