  Let $X$ be a projective smooth geometrically connected curve defined over a
finite field $\mathbb{F}_q$ of cardinality $q$. Let $S$ be a finite set of
closed points of $X$. Let $\bar{X}$ and $\bar{S}$ be the base change of $X$,
$S$ to an algebraic closure. We consider the set of $\ell$-adic ($\ell\nmid q$)
local systems of rank $n$ over $\bar{X}-\bar{S}$ with prescribed tame regular
semisimple and generic ramifications in $\bar{S}$. The genericity ensures that
such an $\ell$-adic local system is automatically irreducible. We show that the
number of these $\ell$-adic local systems fixed by Frobenius endomorphism
equals the number of stable logarithmic Higgs bundles of rank $n$ and degree
$e$ coprime to $n$, with a fixed residue, up to a power of $q$. In the split
case, this number is equal to the number of stable parabolic Higgs bundles
(with full flag structures) fixed by $\mathbb{G}_m$-action with generic
parabolic weights.
