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Lambert series
From Wikipedia, the free encyclopedia
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:
-
bm = (a * 1)(m) = ∑ an n | m
This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.
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Since this last sum is a typical number-theoretic sum, almost any multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has
where σ0(n) = d(n) is the number of positive divisors of the number n.
For the higher order sigma functions, one has
where α is any complex number and
-
σα(n) = (Idα * 1)(n) = ∑ dα d | n
is the divisor function.
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.
Other Lambert series include those for the Mobius function μ(n):
For Euler's totient function φ(n):
For Liouville's function λ(n):
with the sum on the left similar to the Ramanujan theta function.
Substituting q = e − z one obtains another common form for the series, as
where
-
bm = (a * 1)(m) = ∑ an n | m
as before. Examples of Lambert series in this form, with z = 2π, occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details.
- Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9
- Eric W. Weisstein, Lambert Series at MathWorld.