This is a non-regular summation method attributed to A.E.Ingham [J. London Math. Soc. 20, 171-180 (1945; Zbl 061.12802)] although published earlier by A.Wintner [Eratosthenian Avergaes, (1943; Zbl 060.10503)]. G.H.Hardy observed [Divergent Series (1949; Zbl 032.05801)] 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 ––> 0 more slowly than the given sequence.
Reviewer:  J.P.Tull
Classif.:  * 40G99 Special methods of summability
                   11N05 Distribution of primes

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