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Professor Herbert Gangl's Outputs (22)

On the Cohomology of GL2 and SL2 over Imaginary Quadratic Fields (2024)
Journal Article
Gangl, H., Gunnells, P. E., Hanke, J., & Yasaki, D. (online). On the Cohomology of GL2 and SL2 over Imaginary Quadratic Fields. Experimental Mathematics, https://doi.org/10.1080/10586458.2024.2379797

We report on computations of the cohomology of (Formula presented.) and (Formula presented.), where D < 0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of... Read More about On the Cohomology of GL2 and SL2 over Imaginary Quadratic Fields.

On two conjectures of Sun concerning Apéry-like series (2023)
Journal Article
Charlton, S., Gangl, H., Lai, L., Xu, C., & Zhao, J. (2023). On two conjectures of Sun concerning Apéry-like series. Forum Mathematicum, 35(6), 1533-1547. https://doi.org/10.1515/forum-2022-0325

We prove two conjectural identities of Z.-W. Sun concerning Apéry-like series. One of the series is alternating, whereas the other one is not. Our main strategy is to convert the series and the alternating series to log-sine-cosine and log-sinh-cosh... Read More about On two conjectures of Sun concerning Apéry-like series.

On functional equations for Nielsen polylogarithms (2021)
Journal Article
Charlton, S., Gangl, H., & Radchenko, D. (2021). On functional equations for Nielsen polylogarithms. Communications in Number Theory and Physics, 15(2), 363-454. https://doi.org/10.4310/cntp.2021.v15.n2.a4

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed moduloLi5 and products of lower weight functions, the weight 5 Nielsen polylogarithm S3,2 satisfies the dilogarithm five-term relation. We also give some functio... Read More about On functional equations for Nielsen polylogarithms.

Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields (2021)
Journal Article
Burns, D., de Jeu, R., Gangl, H., Rahm, A. D., & Yasaki, D. (2021). Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields. Forum of Mathematics, Sigma, 9, Article e40. https://doi.org/10.1017/fms.2021.9

We develop methods for constructing explicit generators, modulo torsion, of the K₃ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3 -space or on direct calculations in suitable pre-Bloch gr... Read More about Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields.

Clean Single-Valued Polylogarithms (2021)
Journal Article
Charlton, S., Duhr, C., & Gangl, H. (2021). Clean Single-Valued Polylogarithms. Symmetry, integrability and geometry: methods and applications, 17, Article 107. https://doi.org/10.3842/SIGMA.2021.107

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy “clean” functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths on... Read More about Clean Single-Valued Polylogarithms.

On the topological computation of K_4 of the Gaussian and Eisenstein integers (2018)
Journal Article
Gangl, H., Dutour Sikiriˇc, M., Gunnells, P., Hanke, J., Schuermann, A., & Yasaki, D. (2019). On the topological computation of K_4 of the Gaussian and Eisenstein integers. Journal of Homotopy and Related Structures, 14, 281-291. https://doi.org/10.1007/s40062-018-0212-8

In this paper we use topological tools to investigate the structure of the algebraic K-groups K4(R) for R=Z[i] and R=Z[ρ] where i:=−1−−−√ and ρ:=(1+−3−−−√)/2. We exploit the close connection between homology groups of GLn(R) for n≤5 and those of rela... Read More about On the topological computation of K_4 of the Gaussian and Eisenstein integers.

On the cohomology of linear groups over imaginary quadratic fields (2016)
Journal Article
Dutour Sikirić, M., Gangl, H., Gunnells, P. E., Hanke, J., Schürmann, A., & Yasaki, D. (2016). On the cohomology of linear groups over imaginary quadratic fields. Journal of Pure and Applied Algebra, 220(7), 2564-2589. https://doi.org/10.1016/j.jpaa.2015.12.002

Let Γ be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D<0. In this paper we investigate the cohomology of Γ for N=3,4 and for a selection of discriminants: D≥−24 when N=3, and D=−3,−4 when N=4...

Tame kernels and second regulators of number fields and their subfields (2013)
Journal Article
Browkin, J., & Gangl, H. (2013). Tame kernels and second regulators of number fields and their subfields. K-Theory, 12(1), 137-165. https://doi.org/10.1017/is013005031jkt229

Assuming a version of the Lichtenbaum conjecture, we apply Brauer-Kuroda relations between the Dedekind zeta function of a number field and the zeta function of some of its subfields to prove formulas relating the order of the tame kernel of a number... Read More about Tame kernels and second regulators of number fields and their subfields.

Generators and Relations for K_2 O_F (2004)
Journal Article
Belabas, K., & Gangl, H. (2004). Generators and Relations for K_2 O_F. K-Theory, 31(3), 195 - 231. https://doi.org/10.1023/b%3Akthe.0000028979.91416.00

Tate's algorithm for computing K_2 O_F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order---the latter, together with some structural results on the p-... Read More about Generators and Relations for K_2 O_F.

Functional equations for higher logarithms (2003)
Journal Article
Gangl, H. (2003). Functional equations for higher logarithms. Selecta Mathematica (New Series), 9(3), 361 - 377. https://doi.org/10.1007/s00029-003-0312-z

Following earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li_m(z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for... Read More about Functional equations for higher logarithms.

On poly(ana)logs I (2002)
Journal Article
Elbaz-Vincent, P., & Gangl, H. (2002). On poly(ana)logs I. Compositio Mathematica, 130(2), 161-214. https://doi.org/10.1023/a%3A1013757217319

We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the fini... Read More about On poly(ana)logs I.