In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resumed 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. 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 where is the number of positive divisors of the number n. For the higher order sum-of-divisor functions, one has where is any complex number and is the divisor function. In particular, for , the Lambert series one gets is which is (up to the factor of ) the logarithmic derivative of the usual generating function for partition numbers Additional Lambert series related to the previous identity include those for the variants of the Möbius function given below Related Lambert series over the Moebius function include the following identities for any prime : The proof of the first identity above follows from a multi-section (or bisection) identity of these Lambert series generating functions in the following form where we denote to be the Lambert series generating function of the arithmetic function f: The second identity in the previous equations follows from the fact that the coefficients of the left-hand-side sum are given by where the function is the multiplicative identity with respect to the operation of Dirichlet convolution of arithmetic functions. For Euler's totient function : For Von Mangoldt function : For Liouville's function : with the sum on the right similar to the Ramanujan theta function, or Jacobi theta function . Note that Lambert series in which the an are trigonometric functions, for example, an = sin(2n x), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions. Generally speaking, we can extend the previous generating function expansion by letting denote the characteristic function of the powers, , for positive natural numbers and defining the generalized m-Liouville lambda function to be the arithmetic function satisfying . This definition of clearly implies that , which in turn shows that We also have a slightly more generalized Lambert series expansion generating the sum of squares function in the form of In