# 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].

## 41 Citations

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## References

SHOWING 1-10 OF 14 REFERENCES

Riemann Hypothesis for Fp(T )

- Mathematics, Philosophy
- 1996

(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]

- Mathematics
- 1996

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

- Mathematics
- 1996

. (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

- Mathematics
- 1997

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.…