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A Tauberian Theorem and Analogues of the Prime Number Theorem

Published online by Cambridge University Press:  20 November 2018

T. M. K. Davison
T. M. K. Davison*
Affiliation:
University of Waterloo, Waterloo, Ontario
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In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and

1

then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 362 - 367
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Ayoub, R., An introduction to the analytic theory of numbers (Providence, 1963).Google Scholar
2. Hardy, G. H. and Wright, E.M., An introduction to the theory of numbers (3rd éd.; Oxford, 1954).Google Scholar
3. Ingham, A. E., Some Tauberian theorems connected with the prime number theorem, J. London Math. Soc, 20 (1945), 171–180.Google Scholar
4. Landau, E., Handbuch der Lehre von der Verteilung der Primzahlen, I, II (Leipzig, 1909).Google Scholar
5. Widder, D. V., The Laplace transform (Princeton, 1946).Google Scholar