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A method for summing series of numbers. The series
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{{MSC|40C}}
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573501.png" /></td> </tr></table>
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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
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$$
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\sum_{n=1}^\infty a_n
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$$
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is summable by Lambert's method to the number $A$, written ${} = A \ \mathrm{(L)}$ if
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$$
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\lim_{y \searrow 0} F(y) = A
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$$
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where $F(y)$ is a [[Lambert series]] in $\exp(-y)$:
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$$
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F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-\exp(-ny)}
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$$
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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.
  
is summable by Lambert's method to the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573502.png" /> if
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As an example,
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$$
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\sum_{n=0}^\infty \frac{\mu(n)}{n} = 0\ \mathrm{(L)}
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$$
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where $\mu$ is the [[Möbius function]]. Hence if this series converges at all, it converges to zero.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573503.png" /></td> </tr></table>
 
 
where
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573504.png" /></td> </tr></table>
 
 
and 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573505.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573506.png" /> (cf. [[Cesàro summation methods|Cesàro summation methods]]) to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573507.png" /> implies its summability by the Lambert method to the same sum, and if the series is summable by the Lambert method to the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057350/l0573508.png" />, then it is also summable by the [[Abel summation method|Abel summation method]] to the same sum. Lambert's summation method is regular (see [[Regular summation methods|Regular summation methods]]).
 
  
 
====References====
 
====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></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  J.H. Lambert,  "Anlage zur Architektonik" , '''2''' , Riga  (1771)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  G.H. Hardy,  "Divergent series" , Clarendon Press  (1949)</TD></TR>
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<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>
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<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>
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<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>
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</table>

Latest revision as of 20:09, 19 August 2014

2010 Mathematics Subject Classification: Primary: 40C [MSN][ZBL]

A summation method for summing series of complex numbers which assigns a sum to certain divergent series: it is 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 [1]. The summability of a series by the Cesàro summation method $(C,k)$ for some $k > -1$ (cf. 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 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

[1] J.H. Lambert, "Anlage zur Architektonik" , 2 , Riga (1771)
[2] G.H. Hardy, "Divergent series" , Clarendon Press (1949)
[3] Jacob Korevaar (2004). "Tauberian theory. A century of developments". Grundlehren der Mathematischen Wissenschaften 329. Springer-Verlag (2004). ISBN 3-540-21058-X. p. 18.
[4] 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.
[5] Norbert Wiener "Tauberian theorems". Ann. Of Math. 33 (1932) 1–100. DOI 10.2307/1968102. JSTOR 1968102.
How to Cite This Entry:
Lambert summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lambert_summation_method&oldid=14975
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article