A note on Ingham's summation method
A series \(\Sigma c_n\) is summable (I) to \(A\) if \[ \lim_{x\to\infty} \frac{1}{x} \sum_{n\leq x} \sum_{d/n}dc_d = A. \] This is a non-regular summation method attributed to \textit{A.E.Ingham} [J. London Math. Soc. 20, 171-180 (1945; Zbl 0061.12802)] although published earlier by \textit{A.Wintner} [Eratosthenian Avergaes, (1943; Zbl 0060.10503)]. \textit{G.H.Hardy} observed [Divergent Series (1949; Zbl 0032.05801)] that if \(\Sigma c_n\) is (I)-summable then \(c_n=o(\log\log n)\). The present paper shows that Hardy's result is best possible by constructing, for any given positive sequence converging to zero, a series \(\Sigma a_n\) which is (I)-summable and for which \(a_n/\log\log n\to 0\) more slowly than the given sequence.
scientific article; zbMATH DE number 3572622