30 September 2025

Numeric Computations of Landau's constant

Authors: Matija Derganc, Luka Urbanč

                    @inproceedings{urbanc2025landaus,
  author = {Matija Derganc and Luka Urbanč},
  title = {{Numeric Computations of Landau's Constant}},
  booktitle = {Proceedings of the 11th Student Computing Research Symposium (SCORES'25)},
  year = {2025},
  address = {Ljubljana, Slovenia},
  publisher = {University of Ljubljana, Faculty of Computer and Information Science},
  isbn = {978-961-7059-18-2},
  doi = {10.51939/scores25},
  url = {https://zalozba.fri.uni-lj.si/SCORES2025.pdf}
}
                

What started as a project for the course Programming 2 during the second semester of the second year of university turned into a project that would also be presented to other students and researchers at the Faculty of Computer and Information Science. Due to our shared interest in complex analysis, Matija and I started developing an algorithm based on Robert Rettingers paper titled “On Computable Approximation of Landau’s Constant”. The paper outlines the structure for a theoretically correct algorithm, which takes a single precision input and returns the value of Landau’s constant up to the given precision. While the algorithm is theoretically finite, memory constraints of modern computers make approximations much too difficult to efficiently compute, resulting in mixed results.

Landau’s constant is a constant value describing the supermum of the radii of discs that can be inscribed inside the image $f(\mathbb{D})$ for functions $f$ belonging to a normalized class of holomorphic functions. One can think of the constant as a quantitative version of the fact that non-constant holomorphic functions are open. The value of the constant is unknown although we do know certain upper and lower bounds. If we label the constant as $\lambda$, we know that $$ \frac{1}{2} < \lambda \leq \frac{\Gamma(\frac{1}{3}) \Gamma (\frac{5}{6})}{\Gamma (\frac{1}{6})} = 0.5432\dots, $$ where the upper bound is also conjectured to be its exact value. The constant is also closely tied to Bloch’s Constant, for which R. Rettinger also proposes a finite algorithm for its numeric computations. Both constants are also presented in a mathematical context in Matija’s Bachelors degree paper.

The complete project paper can be found here, while one can also browse through the other work presented at the conference in the Proceedings of the 11th Student Computing Research Symposium.