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

$\underset{r\to 1-}{\lim }(1-r)\sum _{n=1}^{\mathrm{\infty }}\frac{n{a}_n{r}^n}{1-r^n}=A.$

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

Examples

• $\sum _{n=1}^{\mathrm{\infty }}\frac{\mu (n)}{n}=0(\mathrm{L})$, 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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