Let $X$ be a K3 surface. I want to prove that $Pic(X)\simeq H^1(X,\mathcal{O}^*_X)$ is torsion-free.
From D.Huybrechts' lectures on K3 surfaces I read that if $L$ is torsion then the Riemann-Roch formula would imply that $L$ is effective. But then if a section $s$ of $L$ has zeroes then $s^k\in H^0(X,L^k)$ has also zeroes, so no positive power of $L$ can be trivial.
What I am missing is how the Riemann-Roch theorem can imply that if $L$ is torsion then $L$ is effective?