Hardy-Littlewood Tauberian Theorem

Let  and suppose

as . Then

as . This theorem is a step in the proof of the prime number theorem, but has subsequently been superseded by an approach due to Wiener (Hardy 1999, p. 34).

REFERENCES:

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 118-119, 1994.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 34-35, 1999.

Hardy, G. H. and Littlewood, J. E. Quart. J. Math. 46, 215-219, 1915.

Hardy, G. H. and Littlewood, J. E. Acta Math. 41, 119-196, 1918.

Karamata. Math. Z. 32, 319-320, 1930.