Criterion for Deciding Zeta-Like Multizeta Values in Positive Characteristic

ABSTRACT

Multizeta values for were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present article, we give an effective algorithm for determining whether a given multizeta value is zeta-like, namely, those whose ratio with the zeta value of the same weight is rational. Moreover, we state conjectures on certain families of zeta-like multizeta values. We provide the data in support of these guesses.

KEYWORDS:

• multizeta values
• zeta-like
• Carlitz tensor powers
• heights

2000 AMS Subject Classification:

• 1R58
• 11G09
• 11M32
• 11M38

Acknowledgments

The authors would like to thank Jing Yu for encouragement, and for suggesting numerous improvements to this paper. We also thank Chieh-Yu Chang, Huei-Jeng Chen, and Fu-Tsun Wei for answering related questions and useful conversations about this problem. Finally, we thank the referee for suggestions to improve the exposition.

Funding

This research was partially supported by the National Science Council, Taiwan.