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Zeta-like multiple zeta values in positive characteristic

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Abstract

We study multiple zeta values (MZV’s) over the rational function field over \({\mathbb {F}}_q\) which were introduced by Thakur as analogues of classical multiple zeta values of Euler. In this paper we affirmatively solve a conjecture of Lara Rodriguez and Thakur which gives a full list of zeta-like MZV’s of weight at most \(q^2\) and depth 2. Further, we completely determine all zeta-like MZV’s of weight at most \(q^2\) and arbitrary depth. Our method is based on a criterion which is derived from the Anderson–Thakur motivic interpretation of MZV’s and the Anderson–Brownawell–Papanikolas criterion for linear independence in positive characteristic.

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Acknowledgements

We are grateful to the anonymous referee for his careful reading of the manuscript and helpful suggestions.

The second author (T. ND) is funded by Vingroup Joint Stock Company and supported by Vingroup Innovation Foundation (VinIF) under the project code VINIF.2021.DA00030. Both authors (HH. L. and T. ND.) were partially supported by ANR Grant COLOSS ANR-19-CE40-0015-02, CNRS IEA “Arithmetic and Galois extensions of function fields” and the Labex MILYON ANR-10-LABX-0070.

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  1. Normandie Université, Université de Caen Normandie - CNRS, Laboratoire de Mathématiques Nicolas Oresme (LMNO), UMR 6139, 14000, Caen, France

    Huy Hung Le & Tuan Ngo Dac

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  1. Huy Hung Le
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  2. Tuan Ngo Dac
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Correspondence to Tuan Ngo Dac.

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Le, H.H., Ngo Dac, T. Zeta-like multiple zeta values in positive characteristic. Math. Z. 301, 2037–2057 (2022). https://doi.org/10.1007/s00209-022-02970-4

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