Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-freemonic polynomials $D(x) \in F_q[x]$ of degree $d$. Katz and Sarnak showed thatthe moments, over $H_{d,q}$, of the zeta functions associated to the curves$y^2=D(x)$, evaluated at the central point, tend, as $q \to \infty$, to themoments of characteristic polynomials, evaluated at the central point, ofmatrices in $USp(2\lfloor (d-1)/2 \rfloor)$. Using techniques that wereoriginally developed for studying moments of $L$-functions over number fields,Andrade and Keating conjectured an asymptotic formula for the moments for $q$fixed and $d \to \infty$. We provide theoretical and numerical evidence infavour of their conjecture. In some cases we are able to work out exactformulas for the moments and use these to precisely determine the size of theremainder term in the predicted moments.
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