On the Instability of the Riemann Hypothesis over Finite Fields

Details On the Instability of the Riemann Hypothesis over Finite Fields On the Instability of the Riemann Hypothesis over Finite Fields

2010-08-03

arxiv.org/abs/1008.0499v1

Mathematics

Complex Variables

16 pages

Scientific paper

We show that it is possible to approximate the zeta-function of a curve over a finite field by meromorphic functions which satisfy the same functional equation and moreover satisfy (respectively do not satisfy) the analogue of the Riemann hypothesis. In the other direction, it is possible to approximate holomorphic functions by simple manipulations of such a zeta-function. We also consider the value distribution of zeta-functions of function fields over finite fields from the viewpoint of Nevanlinna theory.

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