View source for Lambert summation method

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A [[Summation methods|summation method]] for summing series of complex numbers which assigns a sum to certain [[divergent series]]: it is [[Regular summation methods|regular]] in that it assigns the sum in the usual sense to any convergent series (an [[Abelian theorem]]). The series
$$
\sum_{n=1}^\infty a_n
$$
is summable by Lambert's method to the number $A$, written ${} = A \ \mathrm{(L)}$ if
$$
\lim_{y \searrow 0} F(y) = A
$$
where $F(y) is a [[Lambert series]] in $\exp(-y)$:
$$
F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-\exp(-ny)}
$$
for $y>0$, if the series on the right-hand side converges. The method was proposed by J.H. Lambert [[#References|[1]]]. The summability of a series by the Cesàro summation method $(C,k)$ for some $k > -1$ (cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum $A$ implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum $A$, then it is also summable by the [[Abel summation method|Abel summation method]] to the same sum. 

As an example,
$$
\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0\ \mathrm{(L)} 
$$
where $\mu$ is the [[Möbius function]].  Hence if this series converges at all, it converges to zero.


====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.H. Lambert,   "Anlage zur Architektonik" , '''2''' , Riga  (1771)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,   "Divergent series" , Clarendon Press  (1949)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">Jacob Korevaar (2004). "Tauberian theory. A century of developments". Grundlehren der Mathematischen Wissenschaften '''329'''. Springer-Verlag (2004). ISBN 3-540-21058-X. p. 18.</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">Hugh L. Montgomery; Robert C. Vaughan (2007). "Multiplicative number theory I. Classical theory". Cambridge tracts in advanced mathematics '''97'''. Cambridge: Cambridge Univ. Press (2007). ISBN 0-521-84903-9.  pp. 159–160.</TD></TR> 
<TR><TD valign="top">[5]</TD> <TD valign="top">Norbert Wiener "Tauberian theorems". ''Ann. Of Math. '' '''33''' (1932) 1–100. {{DOI|10.2307/1968102}}. JSTOR 1968102.</TD></TR>
</table>