Hyper-Mahler measures via Goncharov-Deligne cyclotomy
Abstract
The hyper-Mahler measures $m_k( 1+x_1+x_2),k\in\mathbb Z_{>1}$ and $m_k( 1+x_1+x_2+x_3),k\in\mathbb Z_{>1}$ are evaluated in closed form via Goncharov-Deligne periods, namely $\mathbb Q$-linear combinations of multiple polylogarithms at cyclotomic points (complex-valued coordinates that are roots of unity). Some infinite series related to these hyper-Mahler measures are also explicitly represented as Goncharov-Deligne periods of levels $1$, $2$, $ 3$, $4$, $6$, $8$, $10$ and $12$.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.17243
- arXiv:
- arXiv:2210.17243
- Bibcode:
- 2022arXiv221017243Z
- Keywords:
-
- Mathematics - Number Theory;
- High Energy Physics - Theory;
- Mathematics - Combinatorics;
- 11G55;
- 11M32;
- 11R06
- E-Print:
- (v1) i+30 pages, 5 tables. (v2) i+37 pages, 7 tables. Results improved and enriched. Maple and Mathematica worksheets available as ancillary files. (v3) 47 pages, 8 tables. Reformatted and corrected. (v4) 51 pages, 8 tables. Accepted version