In mathematical analysis and analytic number theory, Lambert summation is a summability method for summing infinite series related to Lambert series specially relevant in analytic number theory.

Definition

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Define the Lambert kernel by   with  . Note that   is decreasing as a function of   when  . A sum   is Lambert summable to   if  , written  .

Abelian and Tauberian theorem

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Abelian theorem: If a series is convergent to   then it is Lambert summable to  .

Tauberian theorem: Suppose that   is Lambert summable to  . Then it is Abel summable to  . In particular, if   is Lambert summable to   and   then   converges to  .

The Tauberian theorem was first proven by G. H. Hardy and John Edensor Littlewood but was not independent of number theory, in fact they used a number-theoretic estimate which is somewhat stronger than the prime number theorem itself. The unsatisfactory situation around the Lambert Tauberian theorem was resolved by Norbert Wiener.

Examples

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  •  , where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence   satisfies the Tauberian condition, therefore the Tauberian theorem implies   in the ordinary sense. This is equivalent to the prime number theorem.
  •   where   is von Mangoldt function and   is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to  . This is equivalent to   where   is the second Chebyshev function.

See also

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References

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  • Jacob Korevaar (2004). Tauberian theory. A century of developments. Grundlehren der Mathematischen Wissenschaften. Vol. 329. Springer-Verlag. p. 18. ISBN 3-540-21058-X.
  • Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. pp. 159–160. ISBN 978-0-521-84903-6.
  • Norbert Wiener (1932). "Tauberian theorems". Ann. of Math. 33 (1). The Annals of Mathematics, Vol. 33, No. 1: 1–100. doi:10.2307/1968102. JSTOR 1968102.