Digit permutations revisited

Goss, David

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 693-728.

Nous considérons ici les fonctions $L$ en caractéristique $p$ ainsi que le groupe $S_{\left(q\right)}$ qui se trouve agir comme des symétries de ces fonctions. Nous expliquons diverses actions de $S_{\left(q\right)}$ qui apparaissent naturellement dans la théorie ainsi que les extensions de ces actions. En général de telles extensions semblent hautement arbitraires, mais dans le cas où les zéros sont non-ramifiés, l’extension est unique (et il est raisonnable de s’attendre à l’unicité seulement dans ce cas-là). Avoir des zéros non-ramifiés est le mieux que l’on puisse espérer en caractéristique positive, et semble êtere un avatar de l’hypothèse de Riemann dans ce contexte. Voir Section 8 pour des discussions plus détaillées.

We discuss here characteristic $p$ $L$-series as well as the group $S_{\left(q\right)}$ which appears to act as symmetries of these functions. We explain various actions of $S_{\left(q\right)}$ that arise naturally in the theory as well as extensions of these actions. In general such extensions appear to be highly arbitrary but in the case where the zeroes are unramified, the extension is unique (and it is reasonable to expect it is unique only in this case). Having unramified zeroes is the best one could hope for in finite characteristic and appears to be an avatar of the Riemann hypothesis in this setting; see Section 8 for a more detailed discussion.

Classification : 11M38, 11G09
Mots clés : $L$-series, Riemann hypothesis, digit permutations, measures, divided algebras

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