Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  367.40005
Autor:  Erdös, Paul; Segal, S.L.
Title:  A note on Ingham's summation method. (In English)
Source:  J. Number Theory 10, 95-98 (1978).
Review:  A series \Sigma cn is summable (I) to A if

limx ––> oo 1/x sumn \leq x sumd/ndcd = A.

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 cn is (I)-summable then cn = 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 an which is (I)-summable and for which an/ 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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