$\sum_{n=1}^{\infty} a_n$ converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \ge 1 $ ; then $a_n=0 , \forall n \in \mathbb N $?
Prove that $a_{n}=0$ for all $n$, if $\sum a_{kn}=0$ for all $k\geq 1$