We study the growth rate of the summatory function of the M"obius functionin the context of an algebraic curve over a finite field. Our work shows astrong resemblance to its number field counterpart, which was proved by Ng in2004. We find an expression for a bound of the summatory function, whichbecomes sharp when the zeta zeros of the curve satisfy a certain linearindependence property. Extending a result of Kowalski in 2008, we prove thatmost curves in the family of universal hyperelliptic curves satisfy thisproperty. Then, we consider a certain geometric average of such bound in thisfamily, using Katz and Sarnak's reformulation of the equidistribution theoremof Deligne. Lastly, we study an asymptotic behavior of this average as thefamily gets larger by evaluating the average values of powers of characteristicpolynomials of random unitary symplectic matrices.
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