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A Lambert series, named after Johann Heinrich Lambert, is a series taking the form
- <math>S(q)=\sum_{n=1}^\infty a_n \frac {q^n}{1-q^n}<math>
It can be resummed by expanding the denominator:
- <math>S(q)=\sum_{n=1}^\infty a_n \sum_{k=1}^\infty q^{nk} = \sum_{m=1}^\infty b_m q^m <math>
where the coefficients of the new series are given by the Dirichlet convolution of <math>{a_n}<math> with the constant function <math>1(n)=1<math>
- <math>b_m = (a*1)(m) = \sum_{n|m} a_n<math>
Since this last sum is a typical number-theortic sum, almost any multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has
- <math>\sum_{n=1}^{\infty} q^n \sigma_0(n) = \sum_{n=1}^{\infty} \frac{q^n}{1-q^n}<math>
where <math>\sigma_0(n)=d(n)<math> is the number of positive divisors of the number <math>n<math>.
Lambert series wherein the an are trigonometric functions, for example, an=sin(2n x), are equal to various combinations of the logarithmic derivatives of Jacobi theta functions.
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