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David Goss
Digit permutations revisited
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 693-728, doi: 10.5802/jtnb.998
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Class. Math.: 11M38, 11G09
Keywords: $L$-series, Riemann hypothesis, digit permutations, measures, divided algebras

Résumé - Abstract

We discuss here characteristic $p$ $L$-series as well as the group $S_{(q)}$ which appears to act as symmetries of these functions. We explain various actions of $S_{(q)}$ 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.

Bibliography

[1] Bruno Anglès 2016, private communication
[2] Bruno Anglès, Tuan Ngo Dac & Floric Tavares Ribeiro, Twisted characteristic $p$ zeta functions, J. Number Theory 168 (2016), p. 180-214 Article
[3] Bruno Anglès & Federico Pellarin, Functional identities for $L$-series values in positive characteristic, J. Number Theory 142 (2014), p. 223-251 Article
[4] Gebhard Böckle, The distribution of the zeros of the Goss zeta function for $A=\mathbb{F}_q[x,y]/(y^2+y+x^3+x+1)$, Math. Z. 275 (2013), p. 835-861 Article
[5] Alain Connes, An essay on the Riemann Hypothesis, Open problems in mathematics, Springer, 2016, p. 225–257
[6] Keith Conrad, The digit principle, J. Number Theory 84 (2000), p. 230-257 Article
[7] Vladimir Gershonovich Drinfeld, Elliptic modules, Mat. Sb. 94(136) (1974), p. 594-627
[8] David Goss, Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 35, Springer, 1996
[9] David Goss, Applications of non-Archimedean integration to the $L$-series of $\tau $-sheaves, J. Number Theory 110 (2005), p. 83-113 Article
[10] David Goss, Zeta phenomenology, in Noncommutative geometry, arithmetic, and related topics, Johns Hopkins University Press, 2011, p. 159-182
[11] David Goss, “A local field approach to the Riemann Hypothesis”, https://arxiv.org/abs/1206.2040, 2012
[12] Kurt Mahler, An interpolation series for continuous functions of a $p$-adic variable, J. Reine Angew. Math 199 (1958), p. 23-34
[13] Matthew A. Papanikolas 2013, private correspondence
[14] Federico Pellarin, Values of certain $L$-series in positive characteristic, Ann. Math. 176 (2012), p. 2055-2093 Article
[15] John Riordan, Combinatorial Identities, Robert E. Krieger Publishing Co., 1979
[16] Jean-Pierre Serre, Endomorphismes complètement continus des espaces Banach $p$-adiques, Publ. Math., Inst. Hautes Étud. Sci. 12 (1962), p. 69-85 Article
[17] Jeffrey T. Sheats, The Riemann hypothesis for the Goss zeta function for $\mathbf{F}_q[T]$, J. Number Theory 71 (1998), p. 121-157 Article
[18] Warren M. Sinnott, Dirichlet series in function fields, J. Number Theory 128 (2008), p. 1893-1899 Article
[19] Dinesh S. Thakur, On characteristic $p$ zeta functions, Compos. Math. 99 (1995), p. 231-247
[20] José Felipe Voloch, Differential operators and interpolation series in power series fields, J. Number Theory 71 (1998), p. 106-108 Article
[21] Carl G. Wagner, Interpolation series for continuous functions on $\pi $-adic completions of ${\rm GF}(q,x)$, Acta Arith. 17 (1971), p. 389-406 Article