# Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

```@article{Faifman2009StatisticsOT,
title={Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field},
author={Dmitry Faifman and Ze{\'e}v Rudnick},
journal={Compositio Mathematica},
year={2009},
volume={146},
pages={81 - 101}
}```
• Published 25 March 2008
• Mathematics
• Compositio Mathematica
Abstract We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g…
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