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Concept
Lambert series
Summary
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.
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.
Official source
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