Let $X$ be a complex surface, $F$ a rank 2 vector bundle on $X$, $n$ a positive integer, $S^nF$ the $n$-th symmetric product of $F$, $P$ a rational divisor, and $a$ a rational number such that $naP$ is integral. Suppose $\det(F)=P+N$ with $P\cdot N=0$, $P$ nef, and $N$ effective. Also suppose $h^0(S^nF(-naP))$ and $h^2(S^nF(-naP))$ are both $O(n^2)$ (https://en.wikipedia.org/wiki/Big_O_notation#Formal_definition). In the proof of Proposition 2.2 of this paper (link: https://mathscinet.ams.org/mathscinet-getitem?mr=1272710), there is the following statement:
Riemann-Roch gives $\chi(S^nF(-naP))=\dfrac{n^3}{6}\{ (1-3a+3a^2)P^2+N^2-c_2(F) \}+O(n^2)$
But how does the Riemann-Roch theorem https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem imply the equality?