Suppose that the series $\sum_{n=1}^{\infty} a_n$ of real terms converges absolutely and $\sum _{n=1}^\infty a_{kn}=0 ,\forall k \in \mathbb N $ , then how to prove that $a_n=0 , \forall n \in \mathbb N $ ?
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4Hint: For every $n$ let $A(n)=\sum\limits_{k\geqslant1}a_{kn}$, then $$a_n=\sum_Q(-1)^{|Q|}A\left(n\prod\limits_{p\in Q}p\right)$$ where the sum is on every finite subset $Q$ of the set of prime numbers. – Did Jun 11 '15 at 13:28
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@Did: Could you please elaborate ? – Jun 11 '15 at 14:12
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http://math.stackexchange.com/questions/30774/prove-that-a-n-0-for-all-n-if-sum-a-kn-0-for-all-k-geq-1 – Jun 11 '15 at 14:50
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@SaunDev Sure -- but only if you explain what your problem with this formula is... otherwise what is the point? – Did Jun 11 '15 at 21:24
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@Did: I don't get the derivation itself and what are the consequences ? – Jun 12 '15 at 01:23