Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resummed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. 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 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 sigma functions, one has where α {displaystyle alpha } is any complex number and