{{Short description|Summability method for a class of divergent series}} 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== Define the Lambert kernel by <math>L(x)=\log(1/x)\frac{x}{1-x}</math> with <math>L(1)=1</math>. Note that <math>L(x^n)>0</math> is decreasing as a function of <math>n</math> when <math>0<x<1</math>. A sum <math>\sum_{n=0}^\infty a_n </math> is Lambert summable to <math>A</math> if <math>\lim_{x\to 1^-}\sum_{n=0}^\infty a_n L(x^n)=A</math>, written <math>\sum_{n=0}^\infty a_n=A\,\,(\mathrm{L})</math>.
==Abelian and Tauberian theorem== Abelian theorem: If a series is convergent to <math>A</math> then it is Lambert summable to <math>A</math>.
Tauberian theorem: Suppose that <math>\sum_{n=1}^\infty a_n</math> is Lambert summable to <math>A</math>. Then it is Abel summable to <math>A</math>. In particular, if <math>\sum_{n=0}^\infty a_n</math> is Lambert summable to <math>A</math> and <math>na_n\geq -C</math> then <math>\sum_{n=0}^\infty a_n</math> converges to <math>A</math>.
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==
* <math>\sum_{n=1}^\infty \frac{\mu(n)}{n} = 0 \,(\mathrm{L})</math>, where μ is the Möbius function. Hence if this series converges at all, it converges to zero. Note that the sequence <math> \frac{\mu(n)}{n}</math> satisfies the Tauberian condition, therefore the Tauberian theorem implies <math>\sum_{n=1}^\infty \frac{\mu(n)}{n}=0</math> in the ordinary sense. This is equivalent to the prime number theorem. * <math>\sum_{n=1}^\infty \frac{\Lambda(n)-1}{n}=-2\gamma\,\,(\mathrm{L})</math> where <math>\Lambda</math> is von Mangoldt function and <math>\gamma</math> is Euler's constant. By the Tauberian theorem, the ordinary sum converges and in particular converges to <math>-2\gamma</math>. This is equivalent to <math>\psi(x)\sim x</math> where <math>\psi</math> is the second Chebyshev function.
==See also== * Lambert series * Abel–Plana formula * Abelian and tauberian theorems
==References== * {{cite book | author=Jacob Korevaar | title=Tauberian theory. A century of developments | series=Grundlehren der Mathematischen Wissenschaften | volume=329 | publisher=Springer-Verlag | year=2004 | isbn=3-540-21058-X | pages=18 }} *{{cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) |author2=Robert C. Vaughan |authorlink2=Robert Charles Vaughan (mathematician) | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=978-0-521-84903-6 | pages=159–160 | publisher=Cambridge Univ. Press | location=Cambridge}} *{{cite journal | author=Norbert Wiener | authorlink=Norbert Wiener | title=Tauberian theorems | journal=Ann. of Math. | year=1932 | volume=33 | pages=1–100 | doi=10.2307/1968102 | issue=1 | publisher=The Annals of Mathematics, Vol. 33, No. 1 | jstor=1968102 }}
Category:Series (mathematics) Category:Summability methods
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