Orders of vanishing of zeros of characteristic $p$ zeta function

Details Orders of vanishing of zeros of characteristic $p$ zeta function Orders of vanishing of zeros of characteristic $p$ zeta function

2006-08-30

arxiv.org/abs/math/0608755v1

Mathematics

Number Theory

Rocky Mountain Journal of Mathematics, to appear

Scientific paper

Orders of vanishing of zeros of zeta functions have much arithmetic information encoded in them. For the absolute zeta function, Dinesh Thakur gave sufficient conditions for the order of vanishing of its zeros when the finite field has two elements. Such conditions consider only principal ideals. This result was generalized by Thakur and Diaz-Vargas. Now the conditions involve not only the principal ideals but all the classes of ideals, still in the field of two elements. In this work, we generalize these results to arbitrary finite fields, using similar proofs of Thakur and Diaz-Vargas.

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