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**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 c_{n} is summable (I) to A if **lim**_{x ––> oo} ^{1}/_{x} **sum**_{n \leq x} **sum**_{d/n}dc_{d} = 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 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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