Partial Sums of a Lambert Series

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

A Lambert series has the form . For bounded coefficients , the series converges in the unit disk and most poles lie on the unit circle. These series play an important role in analytic number theory and are related to the divisor function, Jacobi theta functions, the Möbius function, Euler's totient function, the Liouville function, the Ramanujan theta function, and the Riemann zeta function.


This Demonstration shows partial sums to the term of the series when [1]. The coloring function was based on [2]. For computing a generalized series, see [3].


Contributed by: Enrique Zeleny (February 2015)
Based on a program by: Simon Woods
Open content licensed under CC BY-NC-SA




[1] E. Wegert, Visual Complex Functions: An Introduction with Phase Portraits, New York: Birkhäuser, 2012.

[2] Mr. Wizard. "How can I generate this "domain coloring” plot?" Mathematica StackExchange. (Feb 5, 2015)

[3] J. Arndt, "On Computing the Generalized Lambert Series."

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.