Let $E$ be an elliptic curve over the rationals without complex multiplication such that any elliptic curve $mathbb{Q}$-isogenous to $E$ has trivial $mathbb{Q}$-torsion. Koblitz conjectured that the number of primes less than $x$ for which$|E(mathbb{F}_p)|$ is prime is asymptotic to $$C_Efrac{x}{(log{x})^2} $$ for $C_E$ some constant dependent on $E.$ Miri and Murty showed that for infinitely many $p,$ $|E(mathbb{F}_p)|$ has at most 16 prime factors using the lower bound sieve and assuming the Generalized Riemann Hypothesis. This thesis generalizes Koblitz;;s conjectures to a function field setting through Drinfeld modules. Let $phi$ be a Drinfeld module of rank 2, and $mathbb{F}_q$ a finite field with every $mathbb{F}_q[t]$-isogeny having no $mathbb{F}_q[t]$-torsion points and with $ext{End}_{overline{k}}(phi)=mathbb{F}_q[t].$ Furthermore assume that for each monic irreducible $lin mathbb{F}_q[t],$ the extension generated by adjoining the $l$-torsion points of $phi$ to $mathbb{F}_q(t)$ is geometric. Then there exists a positive constant $C_{phi}$ depending on $phi$ such that there are more than $$ C_{phi}frac{q^x}{x^2}$$ monic irreducible polynomials $P$ with degree less then $x$ such that $chi_{phi}(P)$ has at most 13 prime factors. To prove this result we develop the theory of Drinfeld modules and a translation of the lower bound sieve to function fields.
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