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On Ingham's Summation Method

Published online by Cambridge University Press:  20 November 2018

S. L. Segal
S. L. Segal*
Affiliation:
University of Rochester
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Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s if

where as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 97 - 105
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hardy, G. H., Divergent series (Oxford, 1949).Google Scholar
2. Ingham, A. E., Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc, 20 (1945), 171–180.Google Scholar
3. Landau, E., Handbuch derLehre von der Verteilung der Primzahlen (New York, 1953).Google Scholar
4. Pennington, W. B., On Ingham summability and summability by Lambert series, Proc. Camb. Philos. Soc., 51, (1955), 65–80.Google Scholar
5. Rajagopal, C. T., A note on Ingham summability and summability by Lambert series, Proc. Indian Acad. Sci., A42, (1955), 41–50.Google Scholar
6. Rubel, L. A., An Abelian theorem for number-theoretic sums, Acta Arith., 6 (1960), 175–177, Correction Acta Arith., 6 (1961), 523.Google Scholar
7. Wintner, A., Eratosthenian averages (Baltimore, 1943).Google Scholar