Let $1<\alpha<\beta<3/2$. Set $$ S(n)= \sum_{i\neq j,~ i,j>0} [i^\alpha+j^\beta]^{-1}[(i+n)^\alpha+(j+n)^\beta]^{-1}. $$ One can check that $S(n)$ is finite. My question is when $n\rightarrow \infty$, what is the asymptotics of $S(n)$, e.g., if it is asymptotically a power function? If yes, what is the exponent?