# The Riemann Hypothesis for the Goss Zeta Function for Fq[T]☆

@article{Sheats1998TheRH,
title={The Riemann Hypothesis for the Goss Zeta Function for Fq[T]☆},
author={Jeffrey T. Sheats},
journal={Journal of Number Theory},
year={1998},
volume={71},
pages={121-157}
}
Letqbe a power of a primep. We prove an assertion of Carlitz which takesqas a parameter. Diaz-Vargas' proof of the Riemann Hypothesis for the Goss zeta function for Fp[T] depends on his verification of Carlitz's assertion for the specific caseq=p[D-V]. Our proof of the general case allows us to extend Diaz-Vargas' proof to Fq[T].
Riemann Hypothesis for Goss $t$-adic Zeta Function
• Mathematics
• 2014
In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.
Valuations of v-adic Power Sums and Zero Distribution for the Goss v-adic Zeta Function for Fq[t]
We study the valuation at an irreducible polynomial v of the v-adic power sum, for exponent k (or −k), of polynomials of a given degree d in Fq[t], as a sequence in d (or k). Understanding these
VALUATIONS OF V-ADIC POWER SUMS AND ZERO DISTRIBUTION FOR GOSS’ V-ADIC ZETA FOR Fq[t]
• Mathematics
• 2013
We study the valuation at an irreducible polynomial v of the vadic power sum, for exponent k (or −k), of polynomials of a given degree d in Fq [t], as a sequence in d (or k). Understanding these
Zero Distribution of $v-$adic Multiple Zeta Values over $\mathbb{F}_q(t)$
This paper aims to study the zero distribution of $v-$adic multiple zeta values over function fields. We show that the interpolated $v-$adic MZVs at negative integers only vanish at what we call the
Stabilized Values of the Generalized Goss Zeta Function
Abstract After a brief review in the first section of the definitions and basic properties of the Riemann and Goss zeta functions, we begin in Section 2 the analysis of the generalized Goss zeta
NEWTON SLOPES FOR ARTIN-SCHREIER-WITT TOWERS
. We ﬁx a monic polynomial f ( x ) ∈ F q [ x ] over a ﬁnite ﬁeld and consider the Artin-Schreier-Witt tower deﬁned by f ( x ); this is a tower of curves · · · → C m → C m − 1 → · · · → C 0 = A 1 ,
2 Cyclotomic Function Fields and Artin-Schreier Extensions
• Mathematics
• 2021
We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves ”affine” class groups. We show that, in general,
The Impact of the Infinite Primes on the Riemann Hypothesis for Characteristic p Valued L-series
In [12] we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on known results for $$\zeta _{\mathbb{F}_r [\theta ]} (s)$$ and two assumptions that

## References

SHOWING 1-10 OF 14 REFERENCES
Riemann Hypothesis for Fp(T )
(T) at infinity is an analogue of thecompletion R of Q at infinity (=the archimedean prime), and hence thestatement is indeed analogous to the classical Riemann hypothesis aboutzeros being on a
Riemann Hypothesis forFp[T]
It has been shown by Wan [W1] that a version of the Riemann hypothesis for characteristicpvalued zeta functions, due to D. Goss, is satisfied forFq[T], withq=pbeing prime. We present another proof of
On the Riemann Hypothesis for the CharacteristicpZeta Function
. (1.1)This is a well defined element in 0. A polynomial f is said to be monic iff(t) is of the form 1+tg(t). The constant 1 is the only monic polynomialof degree zero. By convention, the constant 1
Basic Structures of Function Field Arithmetic
1. Additive Polynomials.- 1.1. Basic Properties.- 1.2. Classification of Additive Polynomials.- 1.3. The Moore Determinant.- 1.4. The Relationship Between k[x] and k{?}.- 1.5. The p-resultant.- 1.6.