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Posts
What does the Epstein zeta function actually do?
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 💸
Reflections on APS Summit 2025: Quantum Breakthroughs and Epstein Zeta Function
Published:
Just got back from the biggest physics conference of the world, the #APSSummit25 hosted in Los Angeles 😁.
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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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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Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod-Teller-Muto term applied to cuboidal phase transitions
Published in Journal of Chemical Physics, 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." J. Chem. Phys. 163, 094104. https://doi.org/10.1063/5.0276677
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Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics
Published in arXiv, 2025
This paper addresses the efficient computation of power-law-based interaction potentials of homogeneous n-dimensional bodies with an infinite d-dimensional array of copies, including their higher-order derivatives.
Recommended citation: Buchheit, A. A., Busse, J. K., Keßler, T., & Rybakov, F. N. (2025). "Zeta expansion for long-range interactions under periodic boundary conditions with applications to micromagnetics." arXiv preprint arXiv:2509.26274.
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talks
Computation and properties of the Epstein zeta function: Application and numerical challenges
Published:
The Epstein zeta function generalizes the Riemann zeta function to oscillatory lattice sums in higher dimensions. Beyond its numerous applications in pure mathematics, it has recently been identified as a key component in simulating exotic quantum materials. In this work, we derive a compact and efficiently computable representation of the Epstein zeta function and examine its analytical properties across all arguments. We introduce a superexponentially convergent algorithm, including error bounds, for computing the Epstein zeta function in arbitrary dimensions. To facilitate the computation of integrals involving the Epstein zeta function, we decompose it into a power-law singularity and a regularized Epstein zeta function, which is analytic in the first Brillouin zone. We present the first implementation of the Epstein zeta function and its regularization for arbitrary real arguments in EpsteinLib, a high-performance C library with Python bindings, and rigorously benchmark its precision and performance against known formulas, achieving full precision across the entire parameter range. Finally, we apply our library to the computation of Casimir energies in multidimensional geometries.
Computation and properties of the Epstein zeta function: EpsteinLib for precision many-body Physics
Published:
Session: MAR-T47 Room: 262B Time: Thursday, 4:36 PM
Zeta Expansion for Long-Range Interactions under Periodic Boundary Conditions with Applications to Micromagnetics
Published:
Seminar presentation on the efficient computation of power-law-based interaction potentials in micromagnetics under periodic boundary conditions. The talk addresses the challenge of computing infinite lattice sums for dipolar interactions and generalized Riesz power-law potentials in arbitrary cuboidal domains. We present a method that achieves machine precision by complementing direct summation with correction terms, with exponential convergence and negligible additional computational cost compared to the truncated summation scheme. The approach includes a superexponentially convergent algorithm in terms of generalized zeta functions which requires special functions such as incomplete Bessel functions.
Zeta Expansion for Long-Range Interacting Classical and Quantum Lattices
Published:
Large singular sums play a central role in many areas of pure and applied mathematics and arise prominently in the simulation of classical and quantum systems with long-range interactions. In this talk, I present a collection of numerical methods, based on generalized zeta functions and their derivatives, that enable the efficient and precise simulation of long-range interacting lattice models in previously inaccessible parameter regimes. The Epstein zeta function, a higher-dimensional analogue of the Riemann zeta function that describes oscillatory lattice sums, has recently become an indispensable tool in the numerical treatment of long-range interacting many-body systems.
teaching
WS 2019/2020: Introductory Course to University Mathematics
Undergraduate course, University of Duisburg-Essen, 2019
Introductory course to prepare students for University Mathematics at University of Duisburg-Essen for the winter semester 2019/2020.
WS 2020/2021: Introductory Course to University Mathematics
Undergraduate course, University of Duisburg-Essen, 2020
Introductory course to prepare students for University Mathematics at University of Duisburg-Essen for the winter semester 2020/2021.
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.