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Hardy-Littlewood Tauberian Theorem

Let a_n>=0 and suppose

sum_(n=1)^inftya_ne^(-an)∼1/a

as a->0^+. Then

sum_(n<=x)a_n∼x

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 This theorem is a step in the proof of the prime number theorem, but has subsequently been superseded by an approach due to Wiener (Hardy 1999, p. 34).

See also

Tauberian Theorem

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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.

Referenced on Wolfram|Alpha

Hardy-Littlewood Tauberian Theorem

Cite this as:

Weisstein, Eric W. "Hardy-Littlewood Tauberian Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hardy-LittlewoodTauberianTheorem.html

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