# Difference between revisions of "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 | \sum_{n=1}^\infty a_n | ||

$$ | $$ | ||

− | is summable by Lambert's method to the number $A$ if | + | is summable by Lambert's method to the number $A$, written ${} = A \ \mathrm{(L)}$ if |

$$ | $$ | ||

− | \lim_{y \ | + | \lim_{y \searrow 0} F(y) = A |

$$ | $$ | ||

where | where | ||

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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. | 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. | ||

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<TR><TD valign="top">[1]</TD> <TD valign="top"> J.H. Lambert, "Anlage zur Architektonik" , '''2''' , Riga (1771)</TD></TR> | <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">[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> | </table> |

## Revision as of 19:14, 25 August 2013

A [Summation methods|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
$$
F(y) = \sum_{n=1}^\infty a_n \frac{n y \exp(-ny)}{1-exp(-ny)}
$$
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=30240