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Riemann hypothesis for the Goss \(t\)-adic zeta function. (English) Zbl 1377.11101

Let \(A = {\mathbb F}_q[t]\) be the polynomial ring over a finite field \({\mathbb F}_q\) and let \(K = {\mathbb F}_q(t)\) be the field of fractions of \(A\). For a prime \(v\) of \(A\), we let \({\mathbb C}_v\) be the completion of an algebraic closure of \(K_v\), the completion with respect to the valuation \(v.\)
Let \[ s = (x,y)\in S_v := {\mathbb C}_v^{\ast}\times \lim_{\overset{\leftarrow j} {\mathbb Z}}\left((q^{\deg v} - 1)p^j {\mathbb Z}\right). \] The \(v\)-adic Goss zeta function as introduced by D. Goss [in: The arithmetic of function fields. Proceedings of the workshop at the Ohio State University, June 17-26, 1991, Columbus, Ohio (USA). Berlin: Walter de Gruyter. 313–402 (1992; Zbl 0806.11028)], is defined as \[ \zeta_v(s) = \sum_{d=0}^\infty\, x^d \sum_{\overset{a\in {\mathbb F}_q[t],\text{monic}}{\deg(a) = d (v, a) =1}} a^y, \] see [D. Goss, Basic structures of function field arithmetic. Berlin: Springer (1996; Zbl 0874.11004)] or [D. S. Thakur, Function field arithmetic. River Edge, NJ: World Scientific (2004; Zbl 1061.11001)] for motivation and details.
In this paper under review, the authors investigate the Riemann hypothesis for \(\zeta_v(s)\) under the restriction that \(v\) is of degree one. Thus one may assume that \(v=t\). In this case, the statement for the Riemann hypothesis is the following. For a fixed \(y\), all zeros in \(x\) of \(\zeta_v(s)\) are in \(K_{(t)} = {\mathbb F}_q((t))\) the completion of \(K\) at the prime \(v = (t)\) and all zeros are simple zeros. Special cases under certain restrictions were obtained in [D. Wan, J. Number Theory, 58, 196–212 (1996; Zbl 0858.11030; D. Goss, J. Number Theory, 82, 299–322 (2000; Zbl 1032.11036)]. The authors remove the restrictions in known cases (still under the assumption that \(v\) is of degree one prime of \(K\)).

MSC:

11M38 Zeta and \(L\)-functions in characteristic \(p\)
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References:

[1] J. Diaz-Vargas, Riemann hypothesis for \(\mathbb{F}_{q}[t]\) , J. Number Theory 59 (1996), 313-318. · Zbl 0862.11040 · doi:10.1006/jnth.1996.0100
[2] D. Goss, Basic structures of function field arithmetic , Springer-Verlag, New York, 1998. · Zbl 0892.11021
[3] —-, A Riemann hypothesis for characteristic \(p\) \(L\)-functions , J. Number Theory 82 (2000), 299-322. · Zbl 1032.11036 · doi:10.1006/jnth.1999.2495
[4] J. Sheats, The Riemann hypothesis for the Goss zeta function for \(\mathbb{F}_{q}[t]\) , J. Number Theory 71 (1998), 121-157. · Zbl 0918.11030 · doi:10.1006/jnth.1998.2232
[5] D. Thakur, Function field arithmetic , World Scientific, Hackensack, NJ, 2004. · Zbl 1061.11001
[6] —-, Valuations of \(v\)-adic power sums and zero distribution for the Goss \(v\)-adic zeta function for \(\mathbb{F}_{q}[t]\) , J. Integer Seq. 16 (2013), 1-18.
[7] D. Wan, On the Riemann hypothesis for the characteristic \(p\) zeta function , J. Number Theory 58 (1996), 196-212. · Zbl 0858.11030 · doi:10.1006/jnth.1996.0074
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