# Lambert summation

In mathematical analysis, Lambert summation is a summability method for a class of divergent series.

Definition

A series $sum a_n$ is "Lambert summable" to "A", written $sum a_n = A \left(mathrm\left\{L\right\}\right)$, if

:$lim_\left\{r ightarrow 1-\right\} \left(1-r\right) sum_\left\{n=1\right\}^infty frac\left\{n a_n r^n\right\}\left\{1-r^n\right\} = A . ,$

If a series is convergent to "A" then it is Lambert summable to "A" (an Abelian theorem).

Examples

* $sum_\left\{n=0\right\}^infty frac\left\{mu\left(n\right)\right\}\left\{n\right\} = 0 \left(mathrm\left\{L\right\}\right)$, where &mu; is the Möbius function. Hence if this series converges at all, it converges to zero.

ee also

* Lambert series
* Abelian and tauberian theorems

References

*
*cite book | author=Hugh L. Montgomery | authorlink=Hugh Montgomery (mathematician) | coauthors=Robert C. Vaughan | title=Multiplicative number theory I. Classical theory | series=Cambridge tracts in advanced mathematics | volume=97 | year=2007 | isbn=0-521-84903-9 | pages=159-160
*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

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