Systoles of surfaces

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 $\sum a_{n}=A\,(\mathrm {L} )$, if

$\lim _{r\rightarrow 1-}(1-r)\sum _{n=1}^{\infty }{\frac {na_{n}r^{n}}{1-r^{n}}}=A.\,$

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

Examples

• $\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}=0(\mathrm {L} )$, where μ is the Möbius function. Hence if this series converges at all, it converges to zero.

See also

• Lambert series
• Abel–Plana formula
• Abelian and tauberian theorems

References

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