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In mathematics , a Lambert series , named for Johann Heinrich Lambert , is a series taking the form
S
(
q
)
=
∑
n
=
1
∞
a
n
q
n
1
−
q
n
{\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}}
It can be resummed formally by expanding the denominator:
S
(
q
)
=
∑
n
=
1
∞
a
n
∑
k
=
1
∞
q
n
k
=
∑
m
=
1
∞
b
m
q
m
{\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 are given by the Dirichlet convolution of
a
n
{\displaystyle {a_{n}}}
1
(
n
)
=
1
{\displaystyle 1(n)=1}
b
m
=
(
a
∗
1
)
(
m
)
=
∑
n
|
m
a
n
{\displaystyle b_{m}=(a*1)(m)=\sum _{n|m}a_{n}}
Since this last sum is a typical number-theoretic sum, almost any multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has
∑
n
=
1
∞
q
n
σ
0
(
n
)
=
∑
n
=
1
∞
q
n
1
−
q
n
{\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{0}(n)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}
where
σ
0
(
n
)
=
d
(
n
)
{\displaystyle \sigma _{0}(n)=d(n)}
divisors of the number
n
{\displaystyle n}
For the higher order sigma functions , one has
∑
n
=
1
∞
q
n
σ
α
(
n
)
=
∑
n
=
1
∞
n
α
q
n
1
−
q
n
{\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{\alpha }(n)=\sum _{n=1}^{\infty }{\frac {n^{\alpha }q^{n}}{1-q^{n}}}}
where
α
{\displaystyle \alpha }
complex number and
σ
α
(
n
)
=
(
Id
α
∗
1
)
(
n
)
=
∑
d
|
n
d
α
{\displaystyle \sigma _{\alpha }(n)=({\textrm {Id}}_{\alpha }*1)(n)=\sum _{d|n}d^{\alpha }}
is the divisor function.
Lambert series in which the a n are trigonometric functions , for example, a n =sin(2n x ), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions .
