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What does the Epstein zeta function actually do?

6 minute read

Published:

When I first started studying mathematics at university, I was obsessed with popular math problems, such as the Riemann Hypothesis. The conjecture states that all zeros of the Riemann zeta function other than \(-2, -4, -6\), … lie on the critical line \(\operatorname{Re}(\nu) = 1/2\). If proven, this would earn you a million dollars 💸

publications

EpsteinLib: Fast and Efficient Computation of the Epstein Zeta Function

Published in GitHub, 2024

EpsteinLib is a C library and Python package for fast and efficient computation of the Epstein zeta function for arbitrary multidimensional lattices.

Recommended citation: Buchheit, A. A., Busse, J., Gutendorf, R., & Schmitz, J. (2024). EpsteinLib: Fast and Efficient Computation of the Epstein Zeta Function. GitHub. https://github.com/epsteinlib/epsteinlib
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Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib

Published in arXiv, 2024

This paper establishes the Epstein zeta function as a powerful tool in numerical analysis by rigorously investigating its analytical properties and enabling its efficient computation.

Recommended citation: Buchheit, A. A., Busse, J., & Gutendorf, R. (2024). "Computation and properties of the Epstein zeta function with high-performance implementation in EpsteinLib." arXiv preprint arXiv:2412.16317.
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Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod-Teller-Muto term applied to cuboidal phase transitions

Published in arXiv, 2025

This paper provides a rigorous analysis of Bain-type cuboidal lattice transformations, incorporating a general (n,m) Lennard-Jones two-body potential and a long-range repulsive Axilrod-Teller-Muto (ATM) three-body potential.

Recommended citation: Robles-Navarro, A., Cooper, S., Buchheit, A. A., Busse, J. K., Burrows, A., Smits, O., & Schwerdtfeger, P. (2025). "Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod-Teller-Muto term applied to cuboidal phase transitions." arXiv preprint arXiv:2504.07338.
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Epstein zeta method for many-body lattice sums

Published in arXiv, 2025

This paper presents an efficiently computable representation of many-body lattice sums in terms of singular integrals over products of Epstein zeta functions.

Recommended citation: Buchheit, A. A., & Busse, J. K. (2025). "Epstein zeta method for many-body lattice sums." arXiv preprint arXiv:2504.11989.
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teaching

WS 2021/2022: Analysis II Intensive Course

Undergraduate course, Heinrich Heine University Düsseldorf, 2021

Intensive course to prepare for the Analysis II re-examination for the winter semester 2021/2022 at Heinrich Heine University Düsseldorf.

SS 2022: Analysis II Intensive Course

Undergraduate course, Heinrich Heine University Düsseldorf, 2022

Intensive course to prepare for the Analysis II re-examination for the summer semester 2022 at Heinrich Heine University Düsseldorf.