# Lambert summation

In

mathematical analysis ,**Lambert summation**is a summability method for a class ofdivergent series .**Definition**A series $sum\; a\_n$ is "Lambert summable" to "A", written $sum\; a\_n\; =\; A\; (mathrm\{L\})$, if

:$lim\_\{r\; ightarrow\; 1-\}\; (1-r)\; sum\_\{n=1\}^infty\; frac\{n\; a\_n\; r^n\}\{1-r^n\}\; =\; A\; .\; ,$

If a series is convergent to "A" then it is Lambert summable to "A" (an

Abelian theorem ).**Examples*** $sum\_\{n=0\}^infty\; frac\{mu(n)\}\{n\}\; =\; 0\; (mathrm\{L\})$, where μ 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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