Lambert series

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 an with the constant function 1(n) = 1:

b m = ( a ∗ 1 ) ( m ) = ∑ n ∣ m a n . {\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.\,}

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.

Examples Since this last sum is a typical number-theoretic sum, almost any natural 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)} is the number of positive divisors of the number n. For the higher order sum-of-divisor functions, one has

∑ n = 1 ∞ q n σ α ( n ) = ∑ n = 1 ∞ n α q n 1 − q n = ∑ n = 1 ∞ Li − α ⁡ ( q n ) {\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{\alpha }(n)=\sum _{n=1}^{\infty }{\frac {n^{\alpha }q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }\operatorname {Li} _{-\alpha }(q^{n})}

where α {\displaystyle \alpha } is any complex number, Li {\displaystyle \operatorname {Li} } is the polylogarithm, and

σ α ( n ) = ( Id α ∗ 1 ) ( n ) = ∑ d ∣ n d α {\displaystyle \sigma _{\alpha }(n)=({\textrm {Id}}_{\alpha }*1)(n)=\sum _{d\mid n}d^{\alpha }\,}

is the divisor function. In particular, for α = 1 {\displaystyle \alpha =1} , the Lambert series one gets is

q F ′ ( q ) F ( q ) {\displaystyle q{\frac {F'(q)}{F(q)}}}

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