  We study the growth rate of the summatory function of the M\"obius function
in the context of an algebraic curve over a finite field. Our work shows a
strong resemblance to its number field counterpart, which was proved by Ng in
2004. We find an expression for a bound of the summatory function, which
becomes sharp when the zeta zeros of the curve satisfy a certain linear
independence property. Extending a result of Kowalski in 2008, we prove that
most curves in the family of universal hyperelliptic curves satisfy this
property. Then, we consider a certain geometric average of such bound in this
family, using Katz and Sarnak's reformulation of the equidistribution theorem
of Deligne. Lastly, we study an asymptotic behavior of this average as the
family gets larger by evaluating the average values of powers of characteristic
polynomials of random unitary symplectic matrices.
