Lambert series

In mathematics , a Lambert series is a special series . It is named after Johann Heinrich Lambert .

definition

The Lambert series is a series with the shape

${\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}}$.

properties

convergence

For the Lambert series does not converge. For , it converges whenever the series converges. If not converging, then the Lambert series converges for everyone for whom the power series converges ( Konrad Knopp's theorem ). ${\displaystyle |q|=1}$ ${\displaystyle |q|\neq 1}$ ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle \sum _{n=1}^{\infty }a_{n}}$ ${\displaystyle q}$ ${\displaystyle \sum _{n=1}^{\infty }a_{n}q^{n}}$

Lambert series as a power series

The Lambert series can be summed up to by means of an extension ${\displaystyle |q|<1}$

${\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}}$,

where the coefficients of the new series result from Dirichlet convolution of with the constant sequence : ${\displaystyle a_{n}}$ ${\displaystyle 1_{(n)}=1}$

${\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}}$.

Alternative form

If you place , you get another common form of the series: ${\displaystyle q=e^{-z}}$

${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz}}$

in which

${\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}\,}$

is.

Examples of the Lambert series in this form, with , appear in expressions of the Riemann zeta function for odd natural numbers. ${\displaystyle z=2\pi }$

literature

• Eric W. Weisstein : Lambert Series . In: MathWorld (English).
• GM Fichtenholz : Differential and integral calculus II (=  university books for mathematics . Volume 62 ). 6th edition. VEB Deutscher Verlag der Wissenschaften , Berlin 1974, p. 323 .