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On Ingham summability and summability by Lambert series

Published online by Cambridge University Press:  24 October 2008

W. B. Pennington
Affiliation:
Westfield CollegeLondon, N.W. 3

Extract

In his paper ‘Some Tauberian theorems connected with the prime number theorem’, Ingham(10) discusses the method of summation of the series defined by

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1955

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References

REFERENCES

(1)Austin, M. C.On the absolute summability of a Dirichlet series. J. Lond. math. Soc. 27 (1952), 189–98.CrossRefGoogle Scholar
(2)Bosanquet, L. S.On convergence and summability factors in a Dirichlet series. J. Lond. math. Soc. 22 (1947), 190–5.CrossRefGoogle Scholar
(3)Chandrasekharan, K. and Minakshisundaram, S.Typical means (Tata Institute Math. Monographs, no. 1 (Oxford, 1952)).Google Scholar
(4)Hardy, G. H.Note on Lambert series. Proc. Lond. math. Soc. (2), 13 (1914), 192–8.CrossRefGoogle Scholar
(5)Hardy, G. H.Divergent series (Oxford, 1949).Google Scholar
(6)Hardy, G. H. and Littlewood, J. E.On a Tauberian theorem for Lambert's series, and some fundamental theorems in the analytic theory of numbers. Proc. Lond. math. Soc. (2), 19 (1921), 2129.CrossRefGoogle Scholar
(7)Hardy, G. H. and Littlewood, J. E.Notes on the theory of series. XX. On Lambert series. Proc. Lond. math. Soc. (2), 41 (1936), 257–70.CrossRefGoogle Scholar
(8)Hardy, G. H. and Riesz, M.The general theory of Dirichlet's series (Cambridge Math. Tracts, no. 18, 1915, reprinted 1952).Google Scholar
(9)Hardy, G. H. and Wright, E. M.An introduction to the theory of numbers, 2nd ed. (Oxford, 1945).Google Scholar
(10)Ingham, A. E.Some Tauberian theorems connected with the prime number theorem. J. Lond. math. Soc. 20 (1945), 171180.CrossRefGoogle Scholar
(11)Landau, E.Handbuch der Lehre von der Verteilung der Primzahlen (Leipzig, 1909).Google Scholar
(12)Riesz, M.Une méthode de sommation équivalente à la méthode des moyennes arithmétiques. C.R. Acad. Sci., Paris, 152 (1911), 1651–4.Google Scholar
(13)Wiener, N.Tauberian theorems. Ann. Math. (2), 33 (1932), 1100.CrossRefGoogle Scholar
(14)Wiener, N.The Fourier integral and certain of its applications (Cambridge, 1933).Google Scholar
(15)Wigert, S.Sur quelques fonctions arithmétiques. Acta Math., Stockh., 37 (1914), 113–40.CrossRefGoogle Scholar